Heat the oil in a large (at least 4-quart) Dutch oven or flameproof casserole over medium-high heat. Season the short ribs with the salt and pepper. In batches, without crowding, add the short ribs to the pot and cook, turning occasionally, until browned on all sides, about 8 minutes. Using tongs, transfer the ribs to a platter.
Pour off all the fat from the pot leaving just the coating of fat along the bottom. Add the onion, chopped carrot, and celery to the pot and reduce the heat to medium-low. Cover and cook, stirring often, until the vegetables are softened, about 5 minutes. Add the garlic, herbes de Provence, and flour and stir until the garlic gives off its aroma, about 1 minute. Stir in the wine and bring to a boil over high heat, stirring up the browned bits on the bottom of the pan with a wooden spoon. Add the broth, tomatoes, and bay leaf. Return the short ribs, and any juices, to the pot. Add cold water if needed to barely reach the top of the ribs and bring to a boil over high heat.
Cover tightly, and simmer, stirring occasionally to change the position of the ribs, until the meat is falling-off-the-bone tender, about 1 hour. Remove pot from heat. Drain or ladle the gravy (sauce) into a zip lock bag and the oil will sit on the top. Pinch off a tiny cut in the corner of the bag and drain the gravy back into the pot discarding the fat/oil. Add baby carrots, potato, and mushrooms and simmer for another 25 minutes or until potatoes are tender adding more wine (or water) if needed to cover meat and vegetables.
Transfer the short ribs to a deep serving platter and cover with aluminum foil to keep warm. Skim off the fat from the surface of the cooking liquid, and discard the bay leaf. Remove the meat off the bones and cut into small pieces returning the beef back to the pot. Bring to a boil over high heat without cover and cook until the liquid is reduced to a sauce consistency, about 10 minutes (the exact time depends on the size of the pot). Season the sauce with salt and pepper if needed.
Serve hot with buttermilk biscuits or soft warm dinner rolls.