user10950: I've heard it taught by various secular teachers that through mindfulness one can uncover the "true authentic self" because we realise that we are not thought but that we are awareness of thought. Please correct me if I'm mistaken but in my understanding this is not in alignment with Buddhist teachings and a misunderstanding of mindfulness.  We can discover that there is an awareness of thought, emotion etc but are we this awareness? Wouldn't Buddha say that even awareness is not self?  If we are not thought, not awareness, who or what is the observer? Is the observer just more impermanent phenomena?

user-1: I think this question 'who is observing' is a blind alley. If in one moment you have a sense that there is observation, or you reflect 'I am observing' or 'I was observing', so what? And if in the next moment there's no such sense or reflection, what then?
These 'who am I' or 'what is observing' questions are common in some types of Zen, but they're not orthodox Soto Zen. Dogen criticised them, and Menzan Zuiho Osho in his Jijuyu-Zanmai says:

Searching for the subject of seeing and hearing is also useless. The
  harder you look for the subject, the more you will become tired of
  wastefully struggling, since what is seeking and what is being sought
  cannot be separated.

Full text here: https://terebess.hu/zen/menzan.html
I'm not a Zen teacher so I'm reluctant to make categorical statements about Buddhism, but I'd say with a reasonable degree of confidence that you're correct that 'we are awareness of thought' is not what Buddhism teaches.
As to meditation, I don't think it helps us to recognise our true nature per se, but it's probably harder to recognise with a head full of thoughts, and Zen meditation generally involves letting go of idle intellection (it's hard to say what is or isn't a misunderstanding of 'mindfulness' these days because the word has become mainstream and none of us can be sure now what it's being used to mean, so it's probably best avoided if you want to get technical).

user295: A being is made up of the 6 sense bases or the 5 aggregates and arise and an impersonal process governed by Dependent Origination. Notion of self is both a view and perception in beings who have not realised reality as it is.
There is nothing 

which the the being has control over, 
which is forms a solid everlasting core,
which is continuous and persists through existence,
worth identifying as a self,

Awareness is also impersonal. It arises due to:

sense-organ, 
sense- object and 
sense-consciousness

Madhu,piṇḍika Sutta
You have no control over awareness. Say you hear a unpleasant sound. You cannot command: "Stop hearing!". If you hear something pleasant you cannot command: "Do not stop!". Also when say watching something interesting. Memories and thoughts may arise which takes you attention from what you are watching. You cannot command: "Stay with watching only!". This lack of control makes this partly unsatisfactory.
Even the observer is:

impermanent
unsatisfactory
not / non self

Also see: Sutta references which Discuss Self and Not Self under Different Contexts

user-1: 
Please correct me if I'm mistaken but in my understanding this is not in alignment with Buddhist teachings

Yeah, this is not in alignment with Buddhist teachings.

We can discover that there is an awareness of thought, emotion etc but are we this awareness?

Most of us are lead to believe that awareness is a continuous thing. According to Buddhism, there are two mistakes in that belief. The first is believing it's continuous. The second is believing it's a thing.
In general, an awareness is an awareness of something. In Buddhism, this is called consciousness, which is six-fold (one for each sense organ, including the mind as a sense organ). These consciousnesses are bindings between sense and object.
So, some people might believe there's an underlying continuous awareness that recognizes a thought. But in Buddhism, we are taught that the moment of recognition is called a contact (between mind organ and thought, "glued" by the mind-consciousness), and before that, there was a very different awareness that was part of another contact with another object through another sense.
The consciousnesses that arise during these events are also called established consciousnesses. It is through contact that there is touching of samsara. When consciousness is not established (i.e. it does not arise), say, when there's no opportunity or interest in "touching anything" (outside, inside, anywhere), then...

“But, bhikkhus, when one does not intend, and one does not plan, and one does not have a tendency towards anything, no basis exists for the maintenance of consciousness. When there is no basis, there is no support for the establishing of consciousness. When consciousness is unestablished and does not come to growth, there is no descent of name-and-form.
-- SN 12.39
“When that consciousness is unestablished, not coming to growth, nongenerative, sn.iii.54 it is liberated. By being liberated, it is steady; by being steady, it is content; by being content, he is not agitated. Being unagitated, he personally attains Nibbāna. He understands: ‘Destroyed is birth, the holy life has been lived, what had to be done has been done, there is no more for this state of being.’”
-- SN 22.53
“If, bhikkhus, one does not intend, and one does not plan, but one still has a tendency towards something, this becomes a basis for the maintenance of consciousness. When there is a basis, there is a support for the establishing of consciousness…. Such is the origin of this whole mass of suffering.
-- SN 12.40

Bhikkhu Sujato also discusses this topic in his blog.

user362: Things that arise due to reasons cease once the reasons cease.
Who or what is observing is panchaupadhänakkhandha 
Panchaupadhänakkhandha arise due to a reason and cease once the reason ceases. 
The reason for panchaupadhänakkhandha to exist is the liking toward them (chanda). Once the liking toward them ceases to exist, so will panchaupadhänakkhandha cease to exist. This is nibbana. 
The method of removing this liking is to understand that panchaupadhänakkhandha in terms of the Four Noble Truths, thereby practicing the Noble Eightfold Path. 

user151: Nothing is the observer but there is observing. One must practice just seeing what arises and falls naturally without words and concepts that we use to make a different reality. Conceptual reality is human made as opposed to Ultimately reality that just is what it is, moment by moment, in our experiential awareness.
If you program computers perhaps this will help: 
Conceptual reality is to Ultimate reality like
Ruby language is to machine language. 
We cannot find a self when we see things as they are. As a rule of thumb, everything we experience is empty of self. Everything. Even the apparent observer or watcher. 
Not self simply shows us that we are empty of self as the trees and ponds and lakes obviously are empty of self too. To see anatta is to see yourself as just a bunch of impersonal processes.-Metta

user8157: The mind observes. Consciousness observes. Awareness observes. 

I've heard it taught by various secular teachers that through mindfulness one can uncover the "true authentic self"

No. Buddhism does not teach there is a "true authentic self". 

because we realise that we are not thought but that we are awareness
  of thought.

No. Ultimately, there is no "we". Buddhism explains there are only five aggregates, namely: 

Physical body
Feeling/sensation
Perception
Thinking/emotions
Sense consciousness


Please correct me if I'm mistaken but in my understanding this is not in alignment with Buddhist teachings and a misunderstanding of
  mindfulness.

Correct. What various secular teachers are teaching is not in alignment with Buddhist teachings.

We can discover that there is an awareness of thought, emotion etc
  but are we this awareness?

The mind discovers & the mind is aware. Consciousness knows thought, as follows:

Dependent on the intellect & ideas there arises consciousness at the intellect. The meeting of the three is contact. 
If anyone were to say, 'The intellect is the self,' that wouldn't be tenable. The arising & falling away of the intellect are discerned.
  And when its arising & falling away are discerned, it would follow
  that 'My self arises & falls away.' That's why it wouldn't be tenable
  if anyone were to say, 'The intellect is the self.' So the intellect
  is not-self. If anyone were to say, 'Ideas are the self,' that
  wouldn't be tenable... Thus the intellect is not-self and ideas are
  not-self. If anyone were to say, 'Consciousness at the intellect is
  the self,' that wouldn't be tenable... Thus the intellect is not-self,
  ideas are not-self, consciousness at the intellect is not-self. If
  anyone were to say, 'Contact at the intellect is the self,' that
  wouldn't be tenable... Thus the intellect is not-self, ideas are
  not-self, consciousness at the intellect is not-self, contact at the
  intellect is not-self.
MN 148

'Consciousness, consciousness': Thus is it said. To what extent, friend, is it said to be 'consciousness'?
'It cognizes, it cognizes': Thus, friend, it is said to be 'consciousness.' And what does it cognize? It cognizes 'pleasant.' It
  cognizes 'painful.' It cognizes 'neither painful nor pleasant.' 'It
  cognizes, it cognizes': Thus it is said to be 'consciousness.'
Discernment (wisdom) & consciousness, friend: Are these qualities conjoined or disjoined? Is it possible, having separated them one from
  the other, to delineate the difference between them?
Discernment (wisdom) & consciousness are conjoined, friend, not disjoined. It's not possible, having separated them one from the other, to
  delineate the difference between them. For what one discerns, that one
  cognizes. What one cognizes, that one discerns. Therefore these
  qualities are conjoined, not disjoined, and it is not possible, having
  separated them one from another, to delineate the difference between
  them.
MN 43



Wouldn't Buddha say that even awareness is not self?

Yes. 

If we are not thought, not awareness, who or what is the observer?

The mind is the observer; awareness is the observer; but "we" is not the observer.

Is the observer just more impermanent phenomena?

Yes. Refer to SN 12.61

It would be better for the uninstructed run-of-the-mill person to hold to the body composed of the four great elements, rather than the
  mind, as the self. Why is that? Because this body composed of the four
  great elements is seen standing for a year, two years, three, four,
  five, ten, twenty, thirty, forty, fifty, a hundred years or more. But
  what's called 'mind,' 'intellect' or 'consciousness' by day and by
  night arises as one thing and ceases as another. Just as a monkey,
  swinging through a forest wilderness, grabs a branch. Letting go of
  it, it grabs another branch. Letting go of that, it grabs another one.
  Letting go of that, it grabs another one. In the same way, what's
  called 'mind,' 'intellect' or 'consciousness' by day and by night
  arises as one thing and ceases as another.
SN 12.61


10950: <p>I've heard it taught by various secular teachers that through mindfulness one can uncover the "true authentic self" because we realise that we are not thought but that we are awareness of thought. Please correct me if I'm mistaken but in my understanding this is not in alignment with Buddhist teachings and a misunderstanding of mindfulness.  We can discover that there is an awareness of thought, emotion etc but are we this awareness? Wouldn't Buddha say that even awareness is not self?  If we are not thought, not awareness, who or what is the observer? Is the observer just more impermanent phenomena?</p>
 -1: <p>I think this question 'who is observing' is a blind alley. If in one moment you have a sense that there is observation, or you reflect 'I am observing' or 'I was observing', so what? And if in the next moment there's no such sense or reflection, what then?</p>

<p>These 'who am I' or 'what is observing' questions are common in some types of Zen, but they're not orthodox Soto Zen. Dogen criticised them, and Menzan Zuiho Osho in his Jijuyu-Zanmai says:</p>

<blockquote>
  <p>Searching for the subject of seeing and hearing is also useless. The
  harder you look for the subject, the more you will become tired of
  wastefully struggling, since what is seeking and what is being sought
  cannot be separated.</p>
</blockquote>

<p>Full text here: <a href="https://terebess.hu/zen/menzan.html" rel="nofollow noreferrer">https://terebess.hu/zen/menzan.html</a></p>

<p>I'm not a Zen teacher so I'm reluctant to make categorical statements about Buddhism, but I'd say with a reasonable degree of confidence that you're correct that 'we are awareness of thought' is not what Buddhism teaches.</p>

<p>As to meditation, I don't think it helps us to recognise our true nature per se, but it's probably harder to recognise with a head full of thoughts, and Zen meditation generally involves letting go of idle intellection (it's hard to say what is or isn't a misunderstanding of 'mindfulness' these days because the word has become mainstream and none of us can be sure now what it's being used to mean, so it's probably best avoided if you want to get technical).</p>
 295: <p>A being is made up of the 6 sense bases or the 5 aggregates and arise and an impersonal process governed by Dependent Origination. Notion of self is both a view and perception in beings who have not realised reality as it is.</p>

<p>There is nothing </p>

<ul>
<li>which the the being has control over, </li>
<li>which is forms a solid everlasting core,</li>
<li>which is continuous and persists through existence,</li>
<li>worth identifying as a self,</li>
</ul>

<p>Awareness is also impersonal. It arises due to:</p>

<ul>
<li>sense-organ, </li>
<li>sense- object and </li>
<li>sense-consciousness</li>
</ul>

<p><a href="http://www.themindingcentre.org/dharmafarer/wp-content/uploads/2009/12/6.14-Madhupindika-S-m18-piya.pdf" rel="nofollow noreferrer">Madhu,piṇḍika Sutta</a></p>

<p>You have no control over awareness. Say you hear a unpleasant sound. You cannot command: "Stop hearing!". If you hear something pleasant you cannot command: "Do not stop!". Also when say watching something interesting. Memories and thoughts may arise which takes you attention from what you are watching. You cannot command: "Stay with watching only!". This lack of control makes this partly unsatisfactory.</p>

<p>Even the observer is:</p>

<ul>
<li>impermanent</li>
<li>unsatisfactory</li>
<li>not / non self</li>
</ul>

<p>Also see: <a href="https://buddhism.stackexchange.com/questions/19165">Sutta references which Discuss Self and Not Self under Different Contexts</a></p>
 -1: <blockquote>
  <p>Please correct me if I'm mistaken but in my understanding this is not in alignment with Buddhist teachings</p>
</blockquote>

<p>Yeah, this is not in alignment with Buddhist teachings.</p>

<blockquote>
  <p>We can discover that there is an awareness of thought, emotion etc but are we this awareness?</p>
</blockquote>

<p>Most of us are lead to believe that awareness is a continuous thing. According to Buddhism, there are two mistakes in that belief. The first is believing it's continuous. The second is believing it's a thing.</p>

<p>In general, an awareness is an awareness <em>of something</em>. In Buddhism, this is called <code>consciousness</code>, which is six-fold (one for each sense organ, including the mind as a sense organ). These consciousnesses are bindings between sense and object.</p>

<p>So, some people might believe there's an underlying continuous awareness that recognizes a thought. But in Buddhism, we are taught that the moment of recognition is called a <code>contact</code> (between mind organ and thought, "glued" by the mind-consciousness), and before that, there was a very different awareness that was part of another contact with another object through another sense.</p>

<p>The consciousnesses that arise during these events are also called <em>established</em> consciousnesses. It is through <code>contact</code> that there is touching of samsara. When consciousness is not established (i.e. it does not arise), say, when there's no opportunity or interest in "touching anything" (outside, inside, anywhere), then...</p>

<blockquote>
  <p>“But, bhikkhus, when one does not intend, and one does not plan, and one does not have a tendency towards anything, no basis exists for the maintenance of consciousness. When there is no basis, there is no support for the establishing of consciousness. When consciousness is unestablished and does not come to growth, there is no descent of name-and-form.</p>
  
  <p>-- <a href="https://suttacentral.net/en/sn12.39" rel="nofollow noreferrer">SN 12.39</a></p>
  
  <p>“When that consciousness is unestablished, not coming to growth, nongenerative, sn.iii.54 it is liberated. By being liberated, it is steady; by being steady, it is content; by being content, he is not agitated. Being unagitated, he personally attains Nibbāna. He understands: ‘Destroyed is birth, the holy life has been lived, what had to be done has been done, there is no more for this state of being.’”</p>
  
  <p>-- <a href="https://suttacentral.net/en/sn22.53" rel="nofollow noreferrer">SN 22.53</a></p>
  
  <p>“If, bhikkhus, one does not intend, and one does not plan, but one still has a tendency towards something, this becomes a basis for the maintenance of consciousness. When there is a basis, there is a support for the establishing of consciousness…. Such is the origin of this whole mass of suffering.</p>
  
  <p>-- <a href="https://suttacentral.net/en/sn12.40" rel="nofollow noreferrer">SN 12.40</a></p>
</blockquote>

<p>Bhikkhu Sujato also discusses this topic in <a href="https://sujato.wordpress.com/2011/05/13/vinna%E1%B9%87a-is-not-nibbana-really-it-just-isn%E2%80%99t/" rel="nofollow noreferrer">his blog</a>.</p>
 362: <p>Things that arise due to reasons cease once the reasons cease.</p>

<p><strong>Who or what is observing is panchaupadhänakkhandha</strong> </p>

<p>Panchaupadhänakkhandha arise due to a reason and cease once the reason ceases. </p>

<p>The reason for panchaupadhänakkhandha to exist is the liking toward them (chanda). Once the liking toward them ceases to exist, so will panchaupadhänakkhandha cease to exist. This is nibbana. </p>

<p>The method of removing this liking is to understand that panchaupadhänakkhandha in terms of the Four Noble Truths, thereby practicing the Noble Eightfold Path. </p>
 151: <p>Nothing is the observer but there is observing. One must practice just seeing what arises and falls naturally without words and concepts that we use to make a different reality. Conceptual reality is human made as opposed to Ultimately reality that just is what it is, moment by moment, in our experiential awareness.</p>

<p>If you program computers perhaps this will help: </p>

<p>Conceptual reality is to Ultimate reality like
Ruby language is to machine language. </p>

<p>We cannot find a self when we see things as they are. As a rule of thumb, everything we experience is empty of self. Everything. Even the apparent observer or watcher. </p>

<p>Not self simply shows us that we are empty of self as the trees and ponds and lakes obviously are empty of self too. To see anatta is to see yourself as just a bunch of impersonal processes.-Metta</p>
 8157: <p>The mind observes. Consciousness observes. Awareness observes. </p>

<blockquote>
  <p><strong><em>I've heard it taught by various secular teachers that through mindfulness one can uncover the "true authentic self"</em></strong></p>
</blockquote>

<p>No. Buddhism does not teach there is a "true authentic self". </p>

<blockquote>
  <p><strong><em>because we realise that we are not thought but that we are awareness
  of thought.</em></strong></p>
</blockquote>

<p>No. Ultimately, there is no "we". Buddhism explains there are only five aggregates, namely: </p>

<ol>
<li>Physical body</li>
<li>Feeling/sensation</li>
<li>Perception</li>
<li>Thinking/emotions</li>
<li>Sense consciousness</li>
</ol>

<blockquote>
  <p><strong><em>Please correct me if I'm mistaken but in my understanding this is not in alignment with Buddhist teachings and a misunderstanding of
  mindfulness.</em></strong></p>
</blockquote>

<p>Correct. What various secular teachers are teaching is not in alignment with Buddhist teachings.</p>

<blockquote>
  <p><strong><em>We can discover that there is an awareness of thought, emotion etc
  but are we this awareness?</em></strong></p>
</blockquote>

<p>The mind discovers &amp; the mind is aware. Consciousness knows thought, as follows:</p>

<blockquote>
  <p><em>Dependent on the intellect &amp; ideas there arises consciousness at the intellect. The meeting of the three is contact.</em> </p>
  
  <p><em>If anyone were to say, 'The intellect is the self,' that wouldn't be tenable. The arising &amp; falling away of the intellect are discerned.
  And when its arising &amp; falling away are discerned, it would follow
  that 'My self arises &amp; falls away.' That's why it wouldn't be tenable
  if anyone were to say, 'The intellect is the self.' So the intellect
  is not-self. If anyone were to say, 'Ideas are the self,' that
  wouldn't be tenable... Thus the intellect is not-self and ideas are
  not-self. If anyone were to say, 'Consciousness at the intellect is
  the self,' that wouldn't be tenable... Thus the intellect is not-self,
  ideas are not-self, consciousness at the intellect is not-self. If
  anyone were to say, 'Contact at the intellect is the self,' that
  wouldn't be tenable... Thus the intellect is not-self, ideas are
  not-self, consciousness at the intellect is not-self, contact at the
  intellect is not-self.</em></p>
  
  <p><em>MN 148</em></p>
  
  <hr>
  
  <p><em>'Consciousness, consciousness': Thus is it said. To what extent, friend, is it said to be 'consciousness'?</em></p>
  
  <p><em>'It cognizes, it cognizes': Thus, friend, it is said to be 'consciousness.' And what does it cognize? It cognizes 'pleasant.' It
  cognizes 'painful.' It cognizes 'neither painful nor pleasant.' 'It
  cognizes, it cognizes': Thus it is said to be 'consciousness.'</em></p>
  
  <p><em>Discernment (wisdom) &amp; consciousness, friend: Are these qualities conjoined or disjoined? Is it possible, having separated them one from
  the other, to delineate the difference between them?</em></p>
  
  <p><em>Discernment (wisdom) &amp; consciousness are conjoined, friend, not disjoined. It's not possible, having separated them one from the other, to
  delineate the difference between them. For what one discerns, that one
  cognizes. What one cognizes, that one discerns. Therefore these
  qualities are conjoined, not disjoined, and it is not possible, having
  separated them one from another, to delineate the difference between
  them.</em></p>
  
  <p><em>MN 43</em></p>
</blockquote>

<hr>

<blockquote>
  <p><strong><em>Wouldn't Buddha say that even awareness is not self?</em></strong></p>
</blockquote>

<p>Yes. </p>

<blockquote>
  <p><strong><em>If we are not thought, not awareness, who or what is the observer?</em></strong></p>
</blockquote>

<p>The mind is the observer; awareness is the observer; but "we" is not the observer.</p>

<blockquote>
  <p><strong><em>Is the observer just more impermanent phenomena?</em></strong></p>
</blockquote>

<p>Yes. Refer to <a href="http://www.accesstoinsight.org/tipitaka/sn/sn12/sn12.061.than.html" rel="nofollow noreferrer">SN 12.61</a></p>

<blockquote>
  <p><em>It would be better for the uninstructed run-of-the-mill person to hold to the body composed of the four great elements, rather than the
  mind, as the self. Why is that? Because this body composed of the four
  great elements is seen standing for a year, two years, three, four,
  five, ten, twenty, thirty, forty, fifty, a hundred years or more. But
  what's called 'mind,' 'intellect' or 'consciousness' by day and by
  night arises as one thing and ceases as another. Just as a monkey,
  swinging through a forest wilderness, grabs a branch. Letting go of
  it, it grabs another branch. Letting go of that, it grabs another one.
  Letting go of that, it grabs another one. In the same way, what's
  called 'mind,' 'intellect' or 'consciousness' by day and by night
  arises as one thing and ceases as another.</em></p>
  
  <p><em>SN 12.61</em></p>
</blockquote>
