Melt 4 tablespoons butter in a small pan. Add shallots, and cook until softened, about 1 1/2 minutes. Stir in water, and season with salt and pepper.
Form 20 patties using 2 tablespoons ground sirloin for each. Place on a rimmed baking sheet, and coat patties with remaining 2 tablespoons butter. Press a pickle slice into each patty (refrigerate for up to 2 hours).
Preheat broiler. Season patties with salt and pepper. Broil for 4 to 5 minutes. Top burgers with cheese during final 30 seconds.
Spread shallot butter on bottom half of each roll, then top with 1 burger. Sandwich with top half of roll, and serve with lettuce, tomatoes, ketchup, mayonnaise, and mustard.