In a medium bowl, mix the meat, onion, garlic, Worcestershire, basil, mustard, egg, bread crumbs, salt, and pepper until just combined, being careful not to over handle the meat.
Shape into 4 patties about 0.75 inch thick. Preheat grill to medium-high.
Grill patties with grill lid up for a minute with the lid up. Then lower lid and grill to desired doneness, flipping once. Top each burger with cheese, and allow to melt. Serve in a toasted bun.