Make sauce: Purée all sauce ingredients in a food processor until very smooth.
Make burgers: In a bowl, combine beef and 1 1/2 tsp. each salt and pepper (if using preformed burgers, sprinkle with salt and pepper). Form into 4 patties about 3/4 in. thick, making a slight depression in each to help keep burgers flat as they cook. Put on a plate, cover, and chill until ready to grill.
Preheat a grill for medium heat (350° to 450°; you can hold your hand 5 in. above cooking grate only 5 to 7 seconds).
Sprinkle onions with remaining salt and pepper and 1 tsp. oil. Grill, covered, until softened, turning once, 8 minutes total.
Meanwhile, rub burgers with 2 tsp. oil and grill, covered, turning once, about 6 minutes total for medium-rare. In the last few moments of cooking, lay a slice of cheese on each burger. Lay bun halves, cut side down, on grill to toast slightly.
Transfer buns to a platter and fill with burgers and onions. Spoon about 1 1/2 tbsp. barbecue sauce on top and add a slice of tomato and a lettuce leaf.
Make ahead: Sauce: 1 week chilled.
Note: Nutritional analysis is per burger, with 1 1/2 tbsp. sauce and trimmings.