> So can a large FPGA be somehow used to brute force encryption?
Yes and no. It is usually possible to use an FPGA to build a specific logic circuit that will run some particular algorithm faster than the algorithm would run on a general-purpose CPU. (By a factor of maybe 2x to 100x. It varies a lot depending on the algorithm.)
It is also the case that if you implement that algorithm on a dedicated custom chip it can always be designed to run faster than the equivalent FPGA.
So if you really want maximum speed, spend the megabucks for dedicated silicon.
That said, modern GPUs and DSP chips might be able to run some algorithms almost as fast as dedicated silicon and faster than FPGAs. So maybe you don't need megabucks. Again, it depends on the algorithm.
In any case you'd need many orders of magnitude of speedup to usefully brute-force a modern crypto algorithm, and the most you'd get even from dedicated silicon is about three. Useful speedup requires quantum computers -- which don't yet exist -- and even quantum computers will only help break asymmetric crypto, not symmetric crypto.