Heat large nonstick skillet over medium heat until hot. Place beef ribeye steaks in skillet; cook steaks 12 to 15 minutes for medium rare (145°F) to medium (160°F) doneness, turning occasionally. Remove steaks; keep warm. Season with salt and pepper, as desired.
Add mushrooms and thyme to same skillet; cook and stir 3 to 5 minutes or until tender. Remove mushrooms; set aside. Add balsamic vinegar to skillet; increase heat to medium-high. Cook and stir 7 to 10 minutes or until browned bits attached to skillet are dissolved and sauce is reduced to 1/4 cup. Stir in butter, mushrooms and 1/4 teaspoon salt. Cook and stir until heated through.
Season steaks with salt and pepper, as desired. Serve sauce with steaks.