Season the short ribs generously with salt and pepper. Place a large cast-iron skillet over high heat and add the oil. Sear the ribs for 1 to 2 minutes per side, working in batches if necessary to prevent crowding.
Place the ribs in a deep pan and add the stock, wine, garlic, carrot, onion, and celery. Make sure the ribs are submerged in the liquid, then cover with foil and braise in the oven for 6 hours, carefully turning them in the liquid halfway through cooking (you want the braised meat to stay attached to the rib bone if possible). Carefully remove the ribs to a plate, cool to room temperature, then refrigerate. The ribs can be braised and cooled the day before frying. (Strain and save the stock for another use—it makes delicious gravy.)
Combine all the seasoned flour ingredients in a large bowl. In a separate bowl, whisk together the eggs and milk to make an egg wash. Dredge the cooled ribs in the seasoned flour, then dip them in the egg wash and dredge again.
Pour 1/2 inch of oil into a Dutch oven or a large, deep-sided cast-iron skillet and clip a frying thermometer to side of the pan. Bring the oil to 325 to 350?F over medium-high heat (the temperature will drop when the ribs are added, so you want to make sure to regulate the heat to maintain a temperature right around 325 to 350?F as you fry). Working in batches, shake off excess flour and carefully add the ribs to the pan, adjusting the heat as needed to ensure the oil maintains an even temperature. Fry each side for 2 to 3 minutes, until golden brown. Drain on a paper towel–lined plate and serve.