Place each steak between two sheets of heavy plastic (or inside a resealable freezer bag) on a solid, level surface. Firmly pound each steak with the smooth side of a meat mallet to a thickness of 1/4 inch. Discard the plastic and generously season steaks with salt and pepper.
Make the barbecue butter sauce: Combine beef broth, barbecue sauce, hot sauce, and another pinch of pepper in a bowl. Add cold butter but do not stir.
Heat oil in a large skillet over high heat until it just begins to smoke, about 1 minute. Place steaks into the pan and sear for 45 to 60 seconds on each side. Remove from the skillet and set aside to rest.
Pour barbecue butter sauce into the skillet and bring to a boil while scraping the browned bits of food off the bottom of the pan with a wooden spoon. Stir occasionally until butter is melted and incorporated, about 2 minutes.