Thanks to Jeanne, I now know the best and most foolproof way to cook filet mignon!
Salt and pepper
Preheat the oven to 400°F. Set a cast iron or oven safe skillet over medium-high heat, add a bit of vegetable oil and preheat for 5 minutes (less for a non-cast iron pan).
Meanwhile, season the steak on both sides with salt and pepper.
When the skillet is ready, add steaks and cook for 2 minutes. Then, flip the steaks and cook for 2 more minutes. Add a pat of butter to each steak then pop the skillet in the oven for 5 minutes.
Remove the steaks to a cutting board, tent with foil and rest for 5 minutes.