In a medium mixing bowl, combine onions, garlic, beef broth, vinegar, thyme and black pepper.
Place steaks in a large zip-lock bag.
Add half of the marinade.
Seal bag and turn over several times to coat the steaks with the marinade.
Refrigerate for 2 hours.
Cover and refrigerate remaining marinade.
Drain and discard marinade from the steaks.
Grill steaks for 5-6 minutes on each side, or until meat reaches desired doneness (145° for medium rare, 160° for medium, 170°for well-done.)
Meanwhile, place remaining marinade in a small saucepan and bring to a boil over medium heat.
Cook and stir for one minute.
Add mushrooms and cook until tender – about 5 minutes.
Serve over steaks.