First, season the short ribs generously with kosher salt. Heat the olive oil in a large Dutch oven or braiser over medium-high heat. Add the ribs, being careful not to crowd the pot, and sear them on all sides until deeply browned.
Remove the seared ribs and set them aside. If there's excess fat, drain some, leaving about two tablespoons. Add the diced onion, celery, and carrot to the pot. Sauté for 3-4 minutes until the onions have softened and become translucent.
Stir in the minced garlic and cook for another minute until fragrant. Add the tomato paste, a teaspoon of salt, and pepper. Cook the paste for 2-3 minutes, stirring constantly. This step is crucial for developing a rich, less acidic tomato flavor.
Pour in the red wine to deglaze the pot, using a wooden spoon to scrape up any browned bits from the bottom. Return the seared ribs to the pot along with the beef broth, crushed tomatoes, herb bundle, and bay leaves.
Bring the mixture to a low simmer. Cover the pot, leaving the lid slightly ajar to allow steam to escape. Let it cook for 2.5 to 3 hours, checking occasionally. The beef is ready when it is fork-tender and falls apart easily.
Remove the herb bundle and bay leaves. Use two forks to shred the beef directly in the pot. Stir in the sherry vinegar to brighten the flavors, and taste for any final seasoning adjustments. Serve hot over your favorite pasta.