Add oil to a heated heavy skillet and sauté round steak, stirring constantly, until browned, about 10 minutes. Remove beef and set aside.
To the same skillet, add onions and mushrooms; sauté until onions are fragrant and translucent, 5 to 7 minutes. Remove and set aside. Return beef to skillet; sprinkle with flour, salt, and pepper. Stir in red wine and beef broth; bring to a boil.
Cover, reduce the heat, and simmer for 1 hour and 15 minutes, or until tender. Add the onion/mushroom mixture and cook until heated thoroughly, another 5 minutes.Servings: 8