Add the oil to a large cast-iron skillet over medium-high heat. Heat until the oil is very hot.
Season the steaks with salt and pepper. Add the steaks to the skillet and sear the steaks on each side for 4 minutes for medium. Remove the steaks to a plate and loosely cover with foil.
Lower the heat to medium and add the mushrooms, shallot, and thyme to the pan and saute until the mushrooms begin to brown and soften, about 5 minutes. You may need to add a bit more oil to the pan. Add the red wine and scrap the bottom of the pan to get the browned bits.
Let the liquid reduce by half, 7 to 8 minutes. Remove from the heat and swirl in 3 tablespoons butter to finish the sauce. Serve the steak with mushroom red wine sauce drizzled over the top.