Pat each filet mignon dry with paper towels, then sprinkle each side with the pepper. Heat a heavy skillet, preferably cast iron, on the stovetop over medium-high heat until a few drops of water dance across the surface. Add the oil, then the steaks, and cook 3 to 4 minutes, adjusting the heat as necessary so that the oil stays hot but does not smoke. Turn the steaks and cook until small drops of red juice come to the surface of the meat, about 5 minutes for medium. Remove to a platter and keep warm.
Increase the heat to high. Add the beef broth to the skillet and scrape up any browned bits from the bottom. Pour in the cognac, if using, and boil 1 to 2 minutes to burn off the alcohol. Remove skillet from heat. Whisk in the butter, 1/2 tablespoon at a time, until melted. Pour the sauce over the steaks.