Theory of Turing machine which outputs Turing machine?
It depends on what your are referring as "theory" (which basically means everything or anything).

Something which is closely related to what you are referring can be the 
field of Program transformations, which tries to study systematically 
programs that modifies other programs. The original interest of this field is the transformers that are semantically invariants (with a bit of hand waving:  you want to study transformers $T$ such that for any program (or a TM) $p$, $T(p)$ does the same thing as $p$). Derives from these original considerations the subfield of program synthesis (here, given some input datas, you want to automatically generate a program that satisfies some conditions encoded by you input).

A good introduction can be the thesis of M. Ward "Proving Program Refinements and Transformations" with an emphasis on the proof of program side.