Preheat oven to 350 degrees F. Dredge ribs in flour, knocking off excess.
Heat bacon fat in a 6-quart Dutch oven over moderately high heat until hot but not smoking. Brown the short ribs. Transfer ribs to a large plate.
Pour off all but about 2 tablespoons bacon fat remaining in Dutch oven, and in it cook garlic, onions, carrots, and rosemary with salt and pepper to taste over moderate heat, stirring until browned lightly; add mushrooms, if desired. Cook mixture about 6 more minutes.
Deglaze the pot with red wine. Add beef broth to the vegetable mixture and bring to a boil, stirring.
Return ribs to the pot and cover. Cook (braise) in the oven until tender, about 2 hours.
