Combine the ground beef, onion, garlic, breadcrumbs, salt, pepper, and egg; mix until thoroughly combined.
Divide the mixture into four equal portions; shape into a patty about 4 inches wide.
Preheat a gas grill to 400˚F.
Arrange the burgers on the grill, and press down the center of each patty with your thumb to make an indentation. Close the lid.
Cook the burgers, turning once, until browned on both sides and no longer pink inside, about 10 minutes.
Top each patty with 1 slice of cheese and continue to cook for 1 minute or until the cheese begins to melt.
Remove from grill and set aside to rest for at least 5 minutes.
Meanwhile, arrange the buns, cut side down, on the grill, and cook until lightly toasted, about 1 minute.
Spread mayonnaise and ketchup on the bottom bun.
Add the remaining cheese slices over the mayo and ketchup. Then, add lettuce, tomato, the burger patty, and sliced red onion.
Top with a bun and serve.