Brush onion with oil. Place patties on grid over medium, ash-covered coals; arrange onion around patties. Grill, uncovered, 13 to 15 minutes (over medium heat on preheated gas grill, covered, 13 to 14 minutes) until instant-read thermometer inserted horizontally into center registers 160°F, turning occasionally. Season with salt and pepper after turning. About 1 minute before patties are done, top each with 2 cheese slices.
Combine mayonnaise and mustard; spread on top halves of rolls. Serve burgers and onion in rolls with lettuce and tomato.