Cut onion crosswise in half. Cut half the onion into 1/4-inch-thick slices; finely chop remaining onion. Mix meat, chopped onions, egg, parsley, garlic and 1 tablespoon mustard just until blended; shape into 4 (1/2-inch-thick) patties.
Brush mushroom and onion slices with dressing. Grill patties and vegetables 6 min., turning vegetables after 3 min. Remove vegetables from barbecue. Turn patties; grill 5 min. (Burgers will not be done.) Meanwhile, cut mushroom into thin slices.
Top burgers with mushrooms, cheese and onions; grill 3 to 4 min. or until burgers are done (160 degrees F) and cheese is melted. Mix remaining mustard and mayonnaise; spread onto buns. Fill with burgers.