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Pat steaks dry and cut into 4 equal portions. Sprinkle with salt and pepper. Heat 1 tablespoon oil in a large skillet over medium-high heat until very hot, but not smoking. Add the steaks and cook until browned on the bottom, 2 to 4 minutes. Turn over, reduce heat to medium-low and cook to desired doneness, 3 to 5 minutes for medium-rare. Remove the steaks from the pan and set aside, covered with foil.
Increase heat to medium-high. Add the remaining 1 tablespoon oil to the pan. Add mushrooms and onion and cook, stirring frequently, until the mushrooms have released their liquid, about 5 minutes. Sprinkle with flour and stir to coat the vegetables, 30 seconds to 1 minute. Pour in broth and wine and cook, stirring and scraping up the browned, until thick enough to coat the back of a spoon, 2 to 4 minutes.
Remove from the heat. Stir in mustard and any accumulated juices from the steak. Serve the steak with about 1/4 cup sauce each.