In large nonstick skillet coated with cooking spray, cook onions, covered, over medium-low heat for 16 to 20 minutes or until onions are very tender, stirring occasionally. Stir in mushrooms and garlic. Cook over medium heat, covered, for 4 to 6 minutes or until onions begin to brown and mushrooms are tender, stirring occasionally. Remove from heat.
Meanwhile, cook veggie burgers according to package directions.
Spread mustard on one side of each bread slice. Place burgers on arugula. Sprinkle with cheese. Spoon onion mixture over burgers. Top with arugula. Serve immediately.