Combine mayonnaise, salt, pepper and cayenne in large bowl with fork. Stir in cheese and roasted peppers. Cover and refrigerate at least 30 minutes.
Meanwhile, shape ground beef into 4 patties. Season, if desired, with salt and black pepper; set aside.
Grill or pan fry burgers, turning once, 8 minutes or until desired doneness. During last 30 seconds of cooking, evenly top each burger with cheese mixture, then cook covered until cheese melts slightly. Arrange burgers on buns, then evenly top with bacon.