Preheat your oven to 160°C (325°F).
Season the lamb shanks generously with salt and pepper on all sides.
In a large oven-safe Dutch oven or heavy-bottomed pot, heat the olive oil over medium-high heat. Once hot, add the lamb shanks and sear until browned on all sides, about 3-4 minutes per side. Remove the shanks from the pot and set aside.
In the same pot, add the diced onion, carrots, and celery. Cook, stirring occasionally, until the vegetables are softened and lightly caramelized, about 5-7 minutes.
Add the minced garlic to the pot and cook for an additional 1 minute until fragrant.
Deglaze the pot with the red wine, scraping up any browned bits from the bottom with a wooden spoon.
Stir in the beef or chicken broth, bay leaves, rosemary, thyme, and tomato paste.
Return the lamb shanks to the pot, nestling them into the liquid and vegetables. Bring the liquid to a simmer.
Cover the pot with a lid and transfer it to the preheated oven. Braise the lamb shanks for 2 to 2 ½ hours, or until the meat is tender and falling off the bone.
Once the lamb shanks are done, remove them from the oven. Discard the bay leaves, rosemary, and thyme sprigs.
Serve the lamb shanks hot, garnished with chopped fresh parsley. Optionally, serve with mashed potatoes, polenta, or crusty bread to soak up the flavorful sauce.