There are many, many different syntactic theories that are Transformational,
Generative, and by Noam Chomsky, and all of them are theoretically intended to
apply to any language, including AAVE. Most of them have not been applied,
however, because there are so many of them and they change so fast.
Syntactic theories like these differ in what assumptions they make about how
humans speak and what are good ways of representing these assumptions in
language descriptions. All of the T, G, by NC theories boil down to certain formal
conventions, most of which are irrelevant for practical language description.
Furthermore, anything dealing with similarities and differences between AAVE and
white-bread American English has to start from the sociolinguistic
parameters of the lects concerned.
So, probably what you should look at first is Sociolinguistic accounts of AAVE and
American English syntactic variations, which are usually stated in some relevant
transformational-generative theory, using only the practical parts. You can pick up
the details you need as you confront the data, which is, after all, the important
thing; theories come and go, but data is data and has to be accounted for.
I'm neither a sociolinguist nor a specialist in TG-NC grammars, so I will leave the
details of what to look for and which stuff to read first to be filled in by my
colleagues here. Until they do, however, google "William Labov", "John Rickford",
"Walt Wolfram", "Dennis Preston", "Penelope Eckert", and "Roger Shuy". All of these
linguists have written extensively and enlighteningly on this and other topics.
Good luck on your research.