Place patties in center of grill over medium, ash-covered coals; arrange onion and bell pepper around patties. Grill, covered, 11 to 13 minutes (over medium heat on preheated gas grill, covered, 7 to 8 minutes), until instant-read thermometer inserted horizontally into center registers 160°F, 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.