Trim the filet mignon steaks of any tough silver skin or gristly bits if needed. If your steak has the chain attached and you do not wish to serve it, cut it away and save for another purpose.
Allow the steaks to rest at room temperature for at least 30 minutes and up to 90 minutes, depending on their thickness. Just before cooking, pat the steaks dry with paper towels. Coat them with oil and salt generously on all sides. This helps create a beautifully flavorful crust during searing. I always recommend letting the steaks sit out to take the chill off for an even cook.
Heat 2 tablespoons of oil in a cast iron pan over high heat for 1 minute, swirling to coat the pan. Place the steaks into the hot pan, then immediately reduce the heat to medium or medium-high depending on your stovetop. Sear the steaks undisturbed for at least 3 minutes or up to 6 minutes per side for a 2-inch thick cut. This initial sear helps develop a beautiful, flavorful crust.
Flip the steaks and cook for another 2 to 5 minutes, adjusting the time based on the thickness and your preferred doneness. For accuracy, use the finger test or an instant-read meat thermometer (120°F to 125°F for rare, 130°F to 135°F for medium rare, 140°F to 145°F for medium). Remove the steaks from the pan, loosely tent them with foil, and let them rest while you prepare the sauce. Resting allows the juices to redistribute—I always make sure to let my steaks rest for at least 5 minutes for maximum juiciness.
Pour the red wine into the same pan and turn the heat to high. Boil the wine vigorously, scraping up any browned bits with a wooden spoon, until reduced by three-quarters, about 10 minutes. Add any juices released by the resting steaks back into the sauce. Once the wine has stopped bubbling, turn off the heat. Whisk in the unsalted butter, one tablespoon at a time, ensuring each addition is fully incorporated before the next. Taste the sauce and adjust with kosher salt as needed. Pour the sauce over the rested steaks and sprinkle them with freshly ground black pepper before serving. For an extra layer of flavor, I like to use a robust red wine—you'll really taste the difference in the sauce!