Preheat the oven: Set your oven temperature to 325°F (163°C).
Brown the lamb shanks: In a large pot, heat 2 tbsp of olive oil over medium-high heat. Season the 4 lamb shanks with salt and pepper. Once the oil is hot, add the lamb shanks to the pot. Brown them on all sides, approximately 3-4 minutes per side. Once browned, remove the lamb shanks and set them aside.
Sauté the vegetables: In the same pot, add the chopped 2 onions, minced 3 cloves of garlic, diced 1 carrot, and diced 2 celery stalks. Sauté for about 5-7 minutes or until the vegetables become softened and fragrant.
Deglaze the pot: Pour in 1 cup of beef broth, using a wooden spoon to scrape up any browned bits from the bottom of the pot.
Mix in additional ingredients: Add the remaining 2 cups of beef broth, the 14 oz can of diced tomatoes, 2 tbsp of tomato paste, 1 tsp of thyme, and 1 tsp of rosemary. Stir thoroughly and let the mixture simmer for about 5 minutes to meld the flavors.
Return the lamb shanks: Place the browned lamb shanks back into the pot. Ensure they are submerged in the liquid. Cover the pot with a lid and transfer it to the preheated oven.
Braise the lamb: Let the lamb shanks braise in the oven for approximately 2 to 2.5 hours or until the meat is fork-tender and falls off the bone.
Adjust seasoning and serve: Once complete, remove the pot from the oven. Taste and adjust the seasoning with more salt and pepper if necessary. Serve hot with your favorite sides.