Preheat your oven to 300°F (150°C).

Season the beef short ribs generously with salt and pepper.

In a large Dutch oven, heat the olive oil over medium-high heat. Sear the short ribs on all sides until browned, about 4-5 minutes per side. Remove the ribs and set aside.

In the same pot, add the diced onion, carrots, and celery. Sauté until softened, about 5-7 minutes. Add the minced garlic and cook for another minute.

Pour in the balsamic vinegar, scraping up any browned bits from the bottom of the pot. Allow it to simmer for about 3 minutes.

Add the beef stock, rosemary, and thyme to the pot. Return the short ribs to the pot, ensuring they are submerged in the liquid.

Cover the pot with a lid and transfer it to the preheated oven. Braise for 3-4 hours, or until the meat is tender and falling off the bone.

Once cooked, remove the short ribs from the pot and set aside. Strain the braising liquid into a saucepan, discarding solids.

In the same saucepan, add the bone marrow and simmer over medium heat until melted. Stir in the butter and season with salt and pepper to taste.

Serve the short ribs topped with the rich bone marrow gravy and garnish with fresh rosemary sprigs.