In a shallow dish large enough to hold the steaks in a single layer, combine the wine, soy sauce, vinegar, thyme, pepper and half of the garlic. Place the steaks in the marinade and turn them over to thoroughly coat. Set them aside to marinate while you prepare the remaining ingredients.
Preheat the broiler or prepare the grill. If broiling, line a broiler pan with foil.
In a medium skillet, melt the butter over medium-high heat. Add the remaining garlic, the scallions and mushrooms and saute until the scallions are limp and the mushrooms are beginning to release their liquid, two to three minutes. Stir in the flour and cook, stirring, until it is no longer visible, about 30 seconds.
Remove the steaks from the marinade, reserving the marinade. Stir the reserved marinade into the skillet and cook, stirring, until the mixture has come to a boil and thickened slightly, one to two minutes. Reduce the heat to low and simmer while you cook the steaks.
Grill or broil the steaks four inches from the heat; 7 minutes per side for rare, 8 minutes per side for medium-rare, 9 minutes per side for well-done. Let the steaks rest for five minutes before serving.
Serve the steaks topped with the red wine mushroom sauce.