In a bowl, mix ground beef, egg, onion, bread crumbs, Worcestershire, garlic, 1/2 teaspoon salt, and 1/4 teaspoon pepper until well blended. Divide mixture into four equal portions and shape each into a patty about 4 inches wide.

Lay burgers on an oiled barbecue grill over a solid bed of hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds); close lid on gas grill. Cook burgers, turning once, until browned on both sides and no longer pink inside (cut to test), 7 to 8 minutes total. Remove from grill.

Lay buns, cut side down, on grill and cook until lightly toasted, 30 seconds to 1 minute.

Spread mayonnaise and ketchup on bun bottoms. Add lettuce, tomato, cheese burger, onion, and salt and pepper to taste. Set bun tops in place.