1. Preheat oven to 400 degrees.
2. In a small bowl, add artichokes, breadcrumbs, egg, and basil and mix together. Add salt and pepper to taste.
3. With a sharp knife, cut a slit through the center of each filet mignon ¬end to end. Push a clean finger into the cut to expand it slightly. Push one-quarter of the stuffing into the hole in each steak.
4. In a cast-iron skillet (or any kind of oven-proof sauté pan), heat oil until smoking. Season all sides of the meat with salt and pepper to taste. Sear the tops of the steaks until browned, about 10 minutes.
5. Turn steaks and place skillet or pan into the oven. Roast meat for 8–12 minutes for medium rare or medium. Roast longer if you desire your meat to be cooked through.
6. Serve with Reese Béarnaise Sauce.