Peel onions and cit into four 1/2 inch thick horizontal slices. Drizzle each slice with 1/2 tsp. balsamic vinegar and 1/2 tsp. olive oil; season to taste with salt.
Grill or pan fry burgers and onions for about 3 minutes per side or til cooked to taste. Top each burger with 1 slice cheese and cook just til cheese melts.
To serve, put each burger on a large lettuce leaf. Add 1 slice onion and barbecue dressing to taste. Top with a smaller lettuce leaf. Bring bottom leaf up and around burger to enclose. Makes 4 servings.
You can substitute large portabello mushrooms for the ground beef patties. Omit the barbecue dressing. instead, drizzle top and bottom of mushrooms with balsamic vinegar, olive oil and soy sauce. Grill or pan fry, then top with onion and wrap in lettuce.