Season short ribs generously with kosher salt.
Heat oil in a large Dutch oven over medium-high heat. Sear short ribs on all sides in batches to avoid crowding. Transfer to a plate.
If excess fat remains, remove all but 2 tbsp from the pot.
Add onion, celery, carrot, and garlic to the pot. Sauté 3-4 minutes until softened.
Stir in tomato paste, 1 tsp kosher salt, and ½ tsp pepper. Cook 2-3 minutes.
Deglaze with red wine, scraping the bottom of the pot.
Return short ribs to the pot. Add broth and crushed tomatoes.
Tie herbs with kitchen twine into a bundle. Add to pot along with bay leaves.
Bring to a low simmer. Cover, leaving lid slightly ajar. Cook 2½–3 hours, until beef is fork-tender.
Remove bay leaves and herb bundle. Shred meat with forks. Discard bones/cartilage if using bone-in ribs.
Stir in vinegar. Simmer uncovered 15–30 mins more to thicken if needed.
Cook pasta according to package. Toss with ragu or spoon over top. Garnish with cheese and parsley.