Preheat an indoor or outdoor grill to medium-high. Place 1 Tbsp. of the oil and the butter in a large skillet and set over medium-high heat. When the butter is melted, add the mushrooms, garlic, lemon juice, Worcestershire sauce, Tabasco sauce, tarragon, salt and pepper. Cook, stirring occasionally, until the mushrooms are tender, about 5 minutes.
While the mushrooms cook, brush the steaks lightly with the remaining oil and season with salt and pepper. Grill the steaks to the desired doneness, allowing, approximately 2 to 3 minutes per side for rare, and 3 to 4 minutes per side for medium-rare. Set a steak on each of 4 plates, top with the mushrooms, and serve.