Combine Ground Beef, onion and 2 tablespoons barbecue sauce in medium bowl, mixing lightly but thoroughly. Shape into four 1/2-inch thick patties.
Place patties on grid over medium, ash-covered coals. Grill, covered, 8 to 10 minutes (over medium heat on preheated gas grill, 7 to 9 minutes) until instant-read thermometer inserted horizontally into center registers 160°F, turning occasionally. About 1 minute before burgers are done, brush with remaining 2 tablespoons barbecue sauce and top with cheese.
Line bottom half of each bun with lettuce leaves, tomato and red onion slices, as desired; top with burger. Close sandwiches.