are you talking about burton bindings?
I wear a size 8 boot and bought a burton small binding (8 being the max for S like your 11 is the max for M). while the boot fit the baseplate and heel cup better than a M, the straps were just too small. even though it was doable, it was a huge pain in the ass strapping in. what i found that worked best for me was getting medium straps and using them on my S binding. perfect fit.
unless you're planning on doing what i did, i would go for the Large.