Lightly shape ground beef into four 1/2-inch thick patties.
Place patties in center of grid over medium, ash-covered coals; arrange onion and bell pepper around patties. 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 160F, turning occasionally. Grill vegetables 13 to 16 minutes or until crisp-tender, turning occasionally. Season with salt and pepper, as desired. Cut bell pepper into 1/2-inch strips.
Spread 1 tablespoon cheese on bottom of each bun; top with burger, another 1 tablespoon cheese, onion slice and 1/4 of bell pepper. Close sandwiches.
Cook's Tip: Cooking times are for fresh or thoroughly thawed ground beef. Color is not a reliable indicator of ground beef doneness.