Season burgers with salt and pepper, set aside. Combine mayonnaise, capers, dill, and lemon juice until just thoroughly incorporated. Set aside. Cook burgers to desired doneness, placing cheese on just at the end so it barely melts. While burgers are cooking, sauté mushrooms in a skillet with a little butter until they are tender, and nicely browned. To build, place mushrooms on bottom side of bun, then place burger pattie on top. Spread caper mayonnaise on top portion of bun. Place lettuce, tomatoes, and pickles to your own desire.