I am wondering if it is possible to have a program that interprets programming code and is capable of turning that code into an optimized circuit. Is this possible, and if so, does it exist?
Yes, it is certainly possible.
There are compilers that will compile from code to circuits. Classic examples would be VHDL or AHDL compilers, but you can certainly do this for any other language, to the extent that you have an upper bound on the running time of the program or that you have an upper bound on the memory use of the program and are willing to have a circuit that contains stateful elements.
In fact, you can walk down to Fry's and buy something that basically has the form you asked for: it's called a computer. That's more or less what a computer does: it takes code written in a programming language and executes it efficiently, using an optimized circuit. Part of the circuit is fixed (it is the gates in the processor) and part of it is data that depends upon the program you are executing (it is the RAM of the computer).
there are different ways to interpret your question but based on comments (incl yours) some of this may be helpful. there is a lot of research in converting higher level programming constructs into FPGAs, field programmable gate arrays, basically hardware where circuits are dynamically reconfigurable. (aka reconfigurable computing).
there is also a special language eg "C for FPGAs", a subset of full C, that is customized for this purpose.
as for getting efficient circuits for particular programs, that is another area of research/ optimization eg circuit optimization/ minimization/ compression. there are also optimization heuristics built as input preprocessors into most SAT solvers. more generally this is known as logic optimization and there are various techniques/ tools for that. also the general theory/ application of optimizing compilers has some overlap with circuit optimization. one simple, near "toy" example or model of a compiler optimization in some cases is short circuit evaluation. the general area you refer to is also an active area of research both theoretical/ applied & there are many related papers on the subject.