Heat grill to medium. In small bowl, stir together mayonnaise and ketchup; set aside.
Divide ground beef into 8 equal portions. Shape 4 into 1/4-inch-thick patties. Shape remaining 4 into 1/2-inch-thick patties, then use thumbs to make shallow 2-inch-wide indent in top of each patty.
Divide cheese among indents (about 1/4 cup each), then top with remaining patties. Firmly pinch edges together (reshape into round patties if needed). Season both sides with a pinch of salt and pepper.
Grill, covered, 3 to 4 minutes per side for medium to medium-well.
Spread buns with mayonnaise mixture, top with burgers, and add toppings as desired.
