01 - Preheat your oven to 325°F (165°C). Season the short ribs generously with salt and pepper. Heat olive oil in a Dutch oven or heavy-bottomed pot over medium-high heat. Sear the short ribs on all sides until deeply browned, approximately 8 minutes total. Remove the short ribs from the pot and set aside.
02 - Add the diced onion, carrots, and celery to the same pot. Sauté until softened, about 5–7 minutes. Stir in the minced garlic and tomato paste, cooking for an additional 1 minute until fragrant.
03 - Pour in the red wine, scraping the bottom of the pot to lift any browned bits. Allow the wine to simmer and reduce by half, which should take about 5 minutes. Add the crushed tomatoes, beef stock, thyme sprigs, rosemary sprig, and bay leaf. Return the seared short ribs to the pot, ensuring they are submerged in the liquid. Bring the mixture to a gentle simmer.
04 - Cover the pot tightly and transfer it to the preheated oven. Braise for 2 hours, or until the meat is fork-tender.
05 - While the short ribs are braising, place the beef marrow bones on a foil-lined baking sheet. Roast them in the oven during the last 20 minutes of the short ribs' braising time, until the marrow is soft and bubbling.
06 - Carefully remove the braised short ribs from the pot. Discard the herb stems and bay leaf. Shred the meat, discarding the bones and any excess fat. Skim any excess fat from the braising liquid. Return the shredded meat to the pot and keep warm over low heat.
07 - Scoop the softened marrow from the roasted bones. Mash the marrow into the room-temperature unsalted butter until well combined. Season with a pinch of salt.
08 - Cook the pappardelle or tagliatelle in a large pot of generously salted boiling water until al dente. Reserve approximately ½ cup of the pasta cooking water before draining the pasta.
09 - Toss the drained pasta with the short rib ragu. Add reserved pasta water as needed, a tablespoon at a time, to achieve a glossy, well-coated consistency.
10 - Remove the pot from the heat. Stir in the prepared bone marrow butter until it melts and coats the pasta and ragu, creating a rich, glossy finish.
11 - Serve the Short Rib Pasta immediately, garnished with chopped fresh parsley and freshly grated Parmesan cheese.