BURGERS: COMBINE all ingredients in bowl. Form into 1/3-lb. patties.
MUSHROOM Topping: Saute onions and mushrooms in butter until caramelized; transfer to bowl. Stir in mustard and A.1.; set aside.
BISTRO Mayo: Whisk all ingredients until well blended; set aside.
FOR each burger: Grill 1 patty to desired doneness. Top with 1 cheese slice.
SPREAD 2 Tbsp. Bistro Mayo onto cut sides of roll. Fill with prepared burger, 2 Tbsp. Mushroom Topping, 1 tomato slice and 1 lettuce leaf.