PREHEAT barbecue to medium-high heat. Reserve 1/4 cup of the dressing; set aside. Brush remaining dressing onto outsides of mushrooms, peppers and one side of the steak.
GRILL mushrooms, peppers and steak, dressing-sides down, 10 min. or until steak is medium doneness (160ºF) and vegetables are crisp-tender, turning after 5 min. Meanwhile, cover four serving plates with greens, then top with onions and cheese.
CUT steak across the grain into thin slices; cut mushrooms and peppers into strips. Top salads with the meat and vegetables. Drizzle with the reserved 1/4 cup dressing.