Combine onions, mushrooms, olive oil, vinegar, soy sauce, brown sugar, garlic, and pepper in a medium bowl. Place steaks in a gallon-sized zip-top bag. Pour other ingredients over steaks. Seal and refrigerate, 8 hours to overnight.

Set an oven rack about 6 inches from the heat source and preheat the oven's broiler. Line a baking sheet with aluminum foil; place steaks on top.

Broil steaks until hot and slightly pink in the center, about 8 minutes per side.

While steaks are broiling, pour onions, mushrooms, and remaining marinade into a saucepan. Simmer over medium-low heat until steaks are cooked. Serve sauce, onions, and mushrooms over steaks.