Combine ground beef, onion, garlic, breadcrumbs, salt, pepper and egg; mix until thoroughly combined.
Divide mixture into four equal portions; shape into a patty about 4 inches wide.
Preheat gas grill.
Lay burgers on gas grill; close lid.
Cook burgers, turning once, until browned on both sides and no longer pink inside, about 7 minutes.
Top with 1 cheese slice and continue to cook for 1 minute, or until cheese begins to melt.
Remove from grill.
Lay buns, cut side down, on grill and cook until lightly toasted, about 1 minute.
Spread mayonnaise and ketchup on bun bottoms.
Top with remaining cheese slices.
Add lettuce, tomato, burger patty, and red onion.
Cover with bun-tops and serve.