Season the short ribs generously with salt and pepper. Heat olive oil in a Dutch oven over medium-high heat. Sear ribs on all sides until deeply browned. Remove and set aside.
Lower heat to medium. Add onion, carrot, celery, and garlic. Sauté 3–5 minutes until softened and fragrant. Deglaze with red wine, scraping up browned bits, and simmer for about 3 minutes.
Add beef broth, tomatoes (crush by hand or spoon), tomato paste, Parmesan rind, salt, pepper, oregano, basil, bay leaves, crushed red pepper, and thyme. Nestle the short ribs back into the sauce.
Bring to a gentle boil, reduce heat, cover, and simmer for 3-4 hours.
Remove bay leaves, thyme stems, and Parmesan rind. Transfer ribs to a plate, discard bones, and shred meat. Return meat to the pot and simmer uncovered for 30 minutes to slightly thicken the sauce. If you want a thicker, more gravy-like texture, let it reduce a bit longer or stir in a splash of pasta water later. Cook pasta in well-salted water until just shy of al dente. Use tongs to transfer pasta directly into the ragù. Toss and cook together for about 5 minutes so the sauce fully coats the noodles.
Spoon into shallow bowls and finish with grated Parmigiano-Reggiano. Optional but encouraged: burrata, flaky salt, and fresh basil.