Season the ribs. Season the beef ribs generously on all sides with 1 Tbsp Kosher salt and ½ tsp freshly ground black pepper. Preheat oven to 350°F (175°C).
Sear the ribs. Preheat a large Dutch oven over high heat, then add a few tablespoons of a light, neutral oil. When very hot, add the ribs. Sear until browned on all sides, about 12-15 minutes total. Transfer seared ribs to a plate, then reduce heat to medium-high.
Sauté the vegetables. At this point you can drain all but 2 Tbsp of fat from the pan if you'd like (but nothing bad will happen if you don't!). Add onion, carrots, and garlic, then cook, stirring occasionally, until translucent, about 3 minutes.
Stir in tomatoes. Add whole tomatoes either by crushing them with your hands directly into the pot or by pouring them in and breaking up with the back of a wooden spoon. Stir in 2 Tbsp tomato paste, then stir until dissolved into sauce.
Add remaining liquid. Pour in 2 cups red wine, 1-2 cups beef or chicken stock, and ¼ cup soy sauce, then bring to a boil.
Cook the ribs. Using kitchen twine, tie together thyme and rosemary, then place in the pot with the bay leaves. Return the seared ribs to the pot, then cover with a lid and transfer to preheated oven. Cook until tender and falling off the bone, about 2½-3 hours.
Shred the beef. Use tongs to carefully remove the beef ribs and place in a large bowl. Cool until you're able to handle the beef with your hands. Discard beef bones, then using your hands or two forks, shred the beef.
Drain the braising liquid. Drain the braising liquid into a serving bowl, discarding the vegetables and herbs. Return the shredded beef to the braising liquid.
Cook the pasta, then serve immediately. When ready to serve, cook pasta until al dente according to package directions, then drain (but reserve about 1 cup of the pasta water!). In a large pan, combine the braised beef and sauce with the cooked pasta, then gently toss over medium heat to thoroughly mix. Feel free to add a splash of reserved pasta water to help thicken, if wanted. Serve immediately with burrata, chopped parsley, and freshly grated Parmesan cheese.