Preheat the oven to 325°F. Pat the short ribs dry, then season generously all over with 5 teaspoons salt and 1 teaspoon pepper.
Heat a large Dutch oven or similar large, sturdy-bottomed, oven-safe pot with a cover (I used a 5.5-quart Dutch oven) over medium-high heat (you don’t need to add any oil). Add about half of the short ribs in a single layer, making sure they are not touching one another (if the pan is too crowded, they won’t brown properly). Sear on all sides until they are deeply browned, about 6 to 8 minutes per batch. Transfer the short ribs to a large baking dish that can catch any juices. Repeat with remaining short ribs.
Carefully discard all but about 2 tablespoons fat from the pot. Reduce the heat to medium and add the carrots, celery, onion, remaining 1 teaspoon salt, and 1/2 teaspoon pepper. Cook, stirring every so often, until the vegetables are tender and beginning to caramelize, about 10 minutes.
Scoot the vegetables to one side to create an open space in the pan. Add the tomato paste and stir vigorously for 30 seconds, then stir the paste into the vegetables to combine. Stir in the garlic and cook 30 seconds.
Add the wine (be careful as it will splatter), and using a wooden spoon or sturdy spatula, scrape up any browned bits that have collected on the pan. Bring the wine to a boil, then reduce the heat to a simmer. Let simmer until the wine is reduced by approximately half, about 10 minutes.
Stir in the stock, tomatoes, Worcestershire, Dijon, thyme bundle, and oregano.
Nestle the short ribs into the pot, along with any cooking liquid that has collected in the pot.
Bring the liquid in the pot to a boil. Cover the pot then transfer to the oven.
Bake the short ribs in the oven, covered, for 1 hour.
Remove the lid, then continue baking, uncovered, for 1 hour more, until the meat is fall-apart tender.
With a slotted spoon, carefully remove the short ribs to a large plate. Check the pot for any bones that have separated from the meat and discard. Fish out and discard the thyme bundle.
Return the pot to the stove and bring the liquid to a simmer. Let simmer for 15 minutes.
With a large spoon, skim off some of the fat from the sauce (you won't be able to get it all; just spoon off what you can). Return the short ribs to the sauce and heat to warm through for 5 minutes. Sprinkle with parsley. Serve hot with mashed potatoes or oven roasted potatoes, papardelle, or cooked polenta.
