Pat the lamb shanks dry with paper towels to ensure a good sear. Heat olive oil in a Dutch oven over medium-high heat until shimmering, then add the shanks. Sear each side until deeply golden brown, about 4-5 minutes per side, until a rich caramel crust forms. Remove and set aside.
Add chopped onions, carrots, and celery to the same pot. Cook over medium heat, stirring occasionally, until the vegetables are softened and golden, about 8 minutes. The kitchen will fill with a sweet aroma as they caramelize.
Stir in the minced garlic and cook for about 1 minute until fragrant, making sure it doesn't burn. Pour in the red wine, scraping up any browned bits from the bottom of the pot with a wooden spoon. Bring to a gentle simmer and cook until the wine reduces by half, about 10 minutes, until the liquid is thick and fragrant.
Return the browned lamb shanks to the pot, nestling them into the wine and vegetable mixture. Add the thyme sprigs, bay leaves, and pour in the broth until the shanks are halfway submerged. Season with salt and pepper to taste.
Cover the pot with a tight-fitting lid and transfer to a preheated oven set to 160°C (320°F). Braise for about 2.5 to 3 hours, or until the meat is falling off the bone and tender. Check occasionally, adding more broth if the liquid level drops too low.
Once cooked, carefully remove the lamb shanks from the pot and tent with foil. Place the pot back on the stove over medium heat to reduce the sauce if needed, simmering until glossy and thick—about 10 minutes. Taste and adjust seasoning as necessary.
Rest the lamb shanks for 10 minutes to allow the juices to redistribute. Then, serve the tender meat topped with the rich, glossy sauce. Pair with crusty bread or mashed potatoes for a comforting, hearty meal.
