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 the oil stays hot but does not smoke.
Turn the steaks and cook until small drops of red juice come to the surface, about 5 minutes for medium. Transfer to a platter and keep warm.
Add the broth to the skillet over high heat and scrape up any browned bits. Pour in the cognac (if desired). Boil 1 to 2 minutes to burn off the alcohol.
Remove the skillet from the heat. Whisk in the butter, one piece at a time, until melted. Pour the sauce over the steaks.