Research in "Research Engineering" from Wikipedia: 


  Software engineering (SE) is the application of a systematic, disciplined,
  quantifiable approach to the development, operation, and maintenance
  of software, and the study of these approaches.


I was wondering if there is something similar for research, something that we could call Research engineering. I imagine it to be a research field on its own, with students "researching on how to do research". I believe software development has benefited a lot from research in SE. Maybe research could also benefit from Research engineering. 

The questions are: 


is there some institute or some university department in the world where they work on Research Engineering?
in which faculty you would position such department/institute?


Edited: After getting a couple of good answers, I am still not completely satisfied, so I would like to clarify my question. What I am really interested in is indeed a "software engineering" approach. I am not interested in philosophical or sociological research. In fact, the question I had originally in mind was whether it's possible or not to apply actual software development methodologies to research. In more concrete terms, I am wondering  whether anyone has studied the application in research of models similar to the waterfall, or the spiral model, or things like extreme programming, Scrum, etc... (Note: these are just examples, please don't comment to each of them one by one).
There is plenty of "research on research" (or "science of science").

There are dividid into different fields, e.g.:


Scientometrics - measuring citations, networks of collaborators, relations between topics and other quantities characterizing the scientific output.
Sociology of science - treating science as a activity of groups of people, with its history etc.
For example there is a great book Ludwik Fleck, Genesis and Development of a Scientific Fact.
Research on collective intelligence, innovation (when, where and how does it happen), etc.
General fields related to education, didactics, teamwork and management.


As you asking about engineering (i.e. how things work in practice), I don't even mention things like philosophy of science. Also, as you see, the answer depends on scale - from an individual, through a group or an institution to a country, the world nowadays or our civilization.

Typically it is done under umbrella of complex systems, complexity, network science, econophysics or data-mining and modeling in sociology. There are institutes doing it, see e.g. the front page of the Santa Fe Institute. Also, there are some projects on it, e.g. QLectives.

ADDED:

As you are interested in the optimization (not only the observation) and on the micro scale: some findings may implicitly give hints, e.g.:


R. Kenna, B. Berche, Critical mass and the dependency of research quality on group size (2010) says that you need a critical number of researchers to make it working smoothly,
B. Jamtveita, E. Jettestuena, J. Mathiesena, Scaling properties of European research units (2009) says that the larger an institute, the higher percentage of administrative workers it has.


Other things may be harder to find, as in science it works a lot in apprentice-master mode, with approaches differing from a group to a group. So it may be not as easy to be serialized (as in different fields, countries, etc. one may need to have different approach); and when you don't a large enough sample, you cannot use quantitive methods in a meaningful way.

Moreover, now we are in the phase preceding formalized studies, as only recently people started to share with the world their soft and subjective findings on that matter, e.g.:


Materials for Nurturing Scientists (a collection) on Uri Alon Lab website (BTW: he is the right person to talk about it, as he promotes such exchange of information on research)
G. Whitesides, Whitesides' Group: Writing a Paper (2004)
J. Gallian, Advice on giving a good powerpoint presentation (2006)


and on things like academia.SE, for a bit of self-reference.