Trim fat from steaks and spread mustard evenly over both sides. In a large skillet heat 1 tablespoon of the oil over medium heat. Add steaks; cook until desired doneness, turning once halfway through cooking (7–10 minutes for medium rare or 10–12 minutes for medium). Transfer to serving platter and keep warm.
Add the reminaing tablespoon oil to drippings in skillet. Add mushrooms; cook and stir for 4 minutes. Stir in wine, Worcestershires sauce, and thyme. Simmer, uncovered, for 3 minutes. Spoon over steaks.