Special equipment: a 3 1/2-inch round cookie cutter
Cut the Cheddar into 8 (1/4-inch-thick by 11/2-inch-wide) slices. Cook the bacon in a large high-sided skillet over medium heat, turning until crisp. Transfer the bacon to a paper towel to drain. Reserve the skillet and 2 tablespoons bacon fat.
Turn the beef out onto a sheet of waxed paper and divide into 16 (3-ounce or generous 1/3 cup) mounds. Press 1 mound into a 3 1/2-inch patty (using the cookie cutter as a guide) and top with some cheese, then put a second mound of beef on top, patting it onto other patty to enclose the cheese. Pinch the edges together to seal and shape into a single patty. Make 7 more burgers in the same manner, then generously season the burgers with salt and pepper.
Heat the reserved bacon fat or oil in a large high-sided skillet over medium heat until hot. Cook the burgers in 2 batches, covered, turning once, about 8 minutes for medium.
Put 2 burgers on the bottom portion of each bun and top with bacon, lettuce, tomato, onion, ketchup, and mustard, if desired. Serve.