| In small bowl, combine sauce ingredients (mayonnaise and Dijon mustard); set aside.
Shape ground beef into four 3/4" thick patties. Lightly brush oil onto cut sides of onion. Place patties and onions on grid over medium, ash covered coals. Grill, uncovered, 13 to 15 minutes or until centers are no longer pink and onions are tender, turning once. Season burgers with salt and pepper, if desired, after turning. Approximately 1 minute before burgers are done, top each burger with 2 slices cheese.
Meanwhile, spread equal parts of sauce mixture on top half of each roll. Line bottom half of each roll with lettuce and tomato; top with cheeseburger and onion. Close sandwiches.