01 - Preheat the oven to 320°F.
02 - Pat the lamb shanks dry and season generously with salt and pepper. Heat olive oil in a large, heavy-based ovenproof pot over medium-high heat. Sear the shanks on all sides until deeply browned. Remove and set aside.
03 - Add diced onion, chopped carrots, and celery to the same pot. Sauté for 5 minutes until softened, then add minced garlic and cook for an additional minute.
04 - Stir in tomato paste and cook for 1 minute to deepen flavor.
05 - Pour in red wine, scraping up browned bits from the bottom. Simmer for 2 to 3 minutes to slightly reduce.
06 - Add beef or lamb stock, chopped tomatoes, rosemary sprigs, thyme sprigs, and bay leaves. Return lamb shanks to the pot, ensuring they are mostly submerged.
07 - Bring to a simmer. Cover with lid and transfer to the preheated oven. Cook for 2 hours 30 minutes to 3 hours, turning shanks once or twice, until meat is very tender.
08 - Thirty minutes before lamb is finished, bring water and salt to a boil in a saucepan. Gradually whisk in polenta. Reduce heat to low and cook, stirring frequently, for 20 to 25 minutes until thick and creamy.
09 - Stir in unsalted butter, grated Parmesan, and chopped rosemary. Adjust seasoning as needed.
10 - Remove lamb shanks from oven. Discard herb stems and bay leaves. If sauce is too thin, simmer uncovered on stovetop to reduce to desired thickness.
11 - Place lamb shanks atop rosemary polenta and spoon sauce over the top before serving.