The ingredients here are the same for any method, but if grilling, there is no pan sauce to be made with garlic and herbs because the steaks are not cooked in a pan. For the oven-only method and grilling method, see my full post.
Arrange an oven rack in the middle position and preheat the oven to 350°F. Place a cast iron skillet (ideal) or a heavy-bottomed stainless steel sauté pan over medium-high heat. To achieve a beautiful sear, let the pan heat up for about 5-8 minutes until it's very hot. A truly hot pan is non-negotiable for that perfect crust. Add a tablespoon of butter and a tablespoon of olive oil and swirl it to coat the entire sauté pan, making sure it shimmers and bubbles before adding the steak.
Place the seasoned filet mignon steak into the hot skillet. You should hear a sizzle immediately. If you don't hear it, remove the steak and continue heating the pan. Sear for 2-3 minutes per side without moving the steak.
Transfer the skillet directly into the preheated oven and bake for 5-8 minutes. Check the internal temperature with a probe thermometer and when it reaches your desired level of doneness (135℉ for medium rare), carefully remove the hot skillet from the oven. Place the steaks on a cutting board or plate and rest 5-10 minutes before slicing.