How do I get a professor at a different institution to read my essay draft? I would like to send a draft of mine to a specific professor. I would be very interested in his comments, however, I am not a student at the moment, so I am not at his institution nor at any other.

Is there any way for me to send him my draft in an email and introduce myself, without sounding rude &amp; incredibly in love with myself? I believe that asking s.o. to read a 5.000 - 10.000 word essay is maybe a bit much? I have no idea how to phrase this or not come across as very demanding and arrogant and I am incredibly concerned.
This is a classic example of what’s been dubbed the “XY problem.” You have some actual problem to which you’ve decided that emailing this professor is the solution and you’re asking about your solution.  But I’m quite confident that getting a professor at another school to read a draft of an essay is not the correct solution to any problem.  You should not be emailing your draft to a professor at another school and no advice we can give on how to email this person is going to make your plan work.  Instead, you should ask a new question about your actual problem.  What kind of essay is this? What kind of problems are you having with writing it?  What kind of help are you looking for?