01 - Set the oven temperature to 320°F.
02 - Pat lamb shanks dry and season thoroughly with salt and black pepper.
03 - Heat olive oil in a large, heavy ovenproof pot over medium-high heat. Brown lamb shanks on all sides then remove and set aside.
04 - In the same pot, add onion, carrots, and celery; sauté for 5 minutes until softened. Add garlic and cook for an additional 1 minute.
05 - Stir in tomato paste and cook for 1 minute to deepen flavor.
06 - Pour in red wine, scraping bottom of the pot to loosen browned bits. Simmer for 2 to 3 minutes to slightly reduce.
07 - Stir in beef or lamb stock, chopped tomatoes, rosemary, thyme, and bay leaves. Return lamb shanks to the pot, submerging them mostly in liquid.
08 - Bring to a simmer, cover with a lid, and transfer to the preheated oven. Cook for 2.5 to 3 hours, turning shanks once or twice until meat is tender and falling off the bone.
09 - Approximately 30 minutes before lamb is done, bring water and salt to a boil in a saucepan. Gradually whisk in polenta, reduce heat to low, and cook while stirring frequently for 20 to 25 minutes until thick and creamy.
10 - Stir butter, Parmesan cheese, and chopped rosemary into the polenta. Adjust seasoning to taste.
11 - Remove lamb shanks from the oven. Discard herb stems and bay leaves. If sauce is too thin, simmer uncovered on the stovetop until reduced to desired consistency.
12 - Plate lamb shanks atop rosemary polenta and spoon sauce over the top.