Remove filets from refrigerator at least 30 minutes before cooking; pat dry with paper towels. Preheat oven to 375°.
In a 12-inch cast-iron skillet, heat oil over medium-high heat. Season both sides of steaks with 3⁄4 teaspoon salt and 1⁄2 teaspoon pepper. Add steaks to skillet. Cook until steaks are browned and can be easily turned, 2 to 3 minutes. Turn steaks.
Place skillet in oven, and bake until a meat thermometer reads 125°, 12 to 15 minutes for medium-rare, or until desired degree of doneness. (Turn steaks once during cooking.) Remove steaks from skillet; cover loosely with foil.
In same skillet, melt butter over medium heat. Add mushrooms and garlic; cook, stirring frequently, until softened, 4 to 5 minutes. In a small bowl, whisk flour into beef broth. Add broth mixture, wine, tomato paste, thyme, and remaining 1⁄4 teaspoon each salt and pepper to skillet. Bring to a boil, stirring frequently. Reduce heat to medium-low; simmer until lightly thickened, 3 to 5 minutes. Return steaks to skillet; cook 1 minute.
Recipe by Southern Cast Iron at https://southerncastiron.com/the-perfect-cast-iron-skillet-steak/