In a large skillet melt the butter over medium heat, add onion slices. Cook uncovered, stirring occasionally, until golden brown. Transfer to a plate and set aside.
Add the vegetable oil to the skillet. Season the liver with salt and pepper. In a shallow dish, dredge liver in flour, shake off excess, reserve the flour. Cook until brown, about 3 minutes on each side. Transfer to a plate.
Return onions to the skillet. Sprinkle with 2 tablespoons of the remaining flour, stir well. Add beef broth or water and bring to a simmer. Return liver to the skillet and cover. Simmer about 12 minutes for medium rare, 15 minutes for medium and 20 minutes for well done. Season the sauce with salt and pepper to taste.