For marinade combine vinegar, Worcestershire sauce, garlic, salt and pepper in small bowl. Gradually whisk in oil until blended.
Combine onion halves, steak and mushrooms in large shallow dish or resealable plastic bag. Add 2/3 cup of marinade; cover or seal and chill 1 hour, turning occasionally. Set aside remaining marinade.
Preheat grill to high. Remove steak and onions from marinade; grill, uncovered, 5 minutes each side or to desired degree of doneness. Remove steak from grill and let stand.
Meanwhile, remove mushrooms from marinade; discard marinade. Grill mushrooms 2 to 3 minutes each side. Cut mushrooms into 1/2-inch-thick slices. Cut steak across grain into thin slices.
Top mixed greens with steak slices, onions and mushrooms. Garnish with tomatoes, if desired. Serve with reserved marinade.
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