Heat the oil in a small pot set over medium-high. Add the mushrooms and onions and cook until tender, about 5 minutes. Mix in flour, tomato paste, garlic and thyme and cook 2 minutes more. Slowly, stirring steadily, mix in the wine. When the mixture is very thick, slowly whisk in the stock. Bring the sauce to a simmer, and simmer until lightly thickened, about 5 minutes. Remove sauce from the heat and set aside until needed.
Preheat your barbecue or indoor grill to medium-high. Brush the steaks with the 1 Tbsp. olive oil; season with salt and pepper.
Set your mushroom sauce back over the heat and return to a simmer. Lightly oil the bars of your grill. Grill the steaks about 3 minutes per side for rare, and 3 to 4 minutes for medium-rare to medium. Plate the steaks and top each one with sauce.