Season the ribs.
Preheat the oven to 325ºF (170ºC). Season the short ribs with salt and pepper. Toss in the flour so they are evenly coated.
Sear the ribs.
Heat a large Dutch oven over medium heat, add the oil and heat. Add short ribs to the pot and sear until browned on all sides, about 6 minutes. Remove to a plate and set aside.
Return the pot to medium heat, add the onion, carrot, celery and garlic, sautéing until the vegetables are tender, about 5 minutes.
Add braising liquid.
Add the tomato paste, red wine, beef stock, rosemary, bay leaf and short ribs (and any remaining juices) to the pot. Bring to a boil then cover the pot and place in the oven.
Braise the short ribs.
Cook, rotating the short ribs about every 45 minutes, until the meat is fork tender, about 2½ to 3 hours.
Add root vegetables to pot.
Add the celery root, potato and onion to a large saucepan. Add the milk, salt and 3 cups of water over the top and bring to a boil over medium-high heat.
Boil root vegetables.
Reduce the heat to medium and simmer until the vegetables are tender easily be pierced with a fork, about 20 to 30 minutes. Drain the vegetables, discarding the cooking liquid.
Blend celery root purée
Add the vegetables to a blender with the butter and Parmesan cheese and blend until pureed. Season to taste with salt.
Assemble and serve.
Spoon celery root puree into bowls and top with braised short ribs. Ladle the braising liquid over the top.