Pat the short ribs with paper towels to remove any excess moisture, then season generously with 2 teaspoons kosher salt and 2 teaspoons pepper.
Add the olive oil to a Dutch oven or large, heavy-bottomed pot and heat over high heat until shimmering. Add half the short ribs and sear on each side until deep brown, about 7 minutes total. Remove from the pan and set aside. Repeat with the remaining short ribs.
Discard all but 1 tablespoon of rendered fat from the pot. Add the onion, carrot, garlic, 1 teaspoon of salt, and 1 teaspoon of pepper. Sauté over medium heat until the vegetables are softened, about 5 minutes.
Add the tomato paste and sauté for a few more minutes, until darkened and thickened.
Pour in the wine and use a wooden spoon to scrape any browned bits, or fond, from the bottom and sides of the pan. Reduce the liquid by half, about 15 minutes.
Add the crushed tomatoes, beef stock, mustard, bay leaves, and thyme to the pot. Stir to combine.
Add the short ribs, making sure the meat is submerged. Bring to a boil, then reduce the heat to low and cover. Simmer for about 2 hours, stirring occasionally. Add up to 1 cup (240 ml) of water as needed if the sauce gets too thick. Skim excess fat off the top as needed. The short ribs are done when the meat pulls apart easily.
Discard any bones that the meat has fallen off of, as well as the bay leaves and thyme stems. Transfer the short ribs to a cutting board. Pull out any remaining bones and discard. Using two forks, shred the meat, then return to the pot.
Add salt and pepper to taste. Simmer for 15 minutes over low heat.
Cook the pasta according to the package instructions. Plate the pasta in shallow bowls and ladle a scoop of ragù on top. Garnish with finely chopped parsley and freshly grated Parmesan cheese.