01 - Preheat the oven to 320°F.
02 - Pat the lamb shanks dry and season generously with salt and black pepper.
03 - Heat olive oil in a large, heavy-based ovenproof pot over medium-high heat and sear lamb shanks on all sides until well browned. Remove and set aside.
04 - Add diced onion, chopped carrots, and celery to the pot and sauté for 5 minutes until softened. Add minced garlic and cook for an additional minute.
05 - Stir in tomato paste and cook for 1 minute to deepen the flavor.
06 - Pour in the red wine, scraping the pot to release browned bits, and simmer for 2 to 3 minutes until slightly reduced.
07 - Add the beef or lamb stock, chopped tomatoes, fresh rosemary, thyme, and bay leaves to the pot. Return lamb shanks to the pot, ensuring they are mostly submerged.
08 - Bring to a simmer, cover, and transfer the pot to the preheated oven. Cook for 2.5 to 3 hours, turning the shanks once or twice, until meat is tender and falling off the bone.
09 - About 30 minutes before the lamb is done, bring water and salt to a boil in a saucepan. Gradually whisk in the polenta, reduce heat to low, and cook while stirring frequently for 20 to 25 minutes until thick and creamy.
10 - Stir in butter, grated Parmesan, and chopped rosemary. Adjust seasoning to taste and keep warm.
11 - Remove lamb shanks from the oven, discard herb stems and bay leaves. If sauce is too thin, simmer uncovered on the stovetop for a few minutes to thicken.
12 - Plate lamb shanks over the rosemary polenta and spoon the sauce generously over the top.