Step 1

Arrange a rack in the middle of the oven and heat to 350°F.

Step 2

Pat the lamb shanks dry with paper towels, then season all over with the pepper and 2 teaspoons of the salt. Heat the oil in a braiser, 6-quart or larger Dutch oven, or wide heavy-bottomed pot over medium-high until shimmering. Working in batches if needed, sear the lamb shanks until browned on the bottom, 4 to 5 minutes. Flip and sear the other side for 4 to 5 minutes more. Transfer the lamb shanks to a large plate.

Step 3

Reduce the heat to medium. Add the onion, carrot, celery, and garlic and cook, stirring occasionally, until softened, about 5 minutes. Sprinkle with the flour, stir to coat the vegetables, and cook 1 minute. Add the wine and scrape the bottom of the pan with a wooden spoon to loosen any browned bits. Simmer until the wine is reduced by half and thickened slightly, about 10 minutes.

Step 4

Stir in the tomatoes and their juices, broth, rosemary, thyme, and remaining 1 teaspoon salt. Bring to a boil, breaking the tomatoes up with the wooden spoon. Return the lamb shanks to the pan and arrange in a single layer, pushing them into the sauce so they’re about halfway covered with liquid.

Step 5

Cover and place in the oven. Cook, flipping the lamb halfway through, until the meat is very tender and pulling away from the bone, 2 to 2 1/2 hours total. Stir in the vinegar, cover again, and set aside to rest on a wire rack for 10 minutes before serving. About 30 minutes before the lamb is ready, make the polenta.

Step 6

Bring the water to a boil in a medium saucepan over medium-high heat. Slowly whisk in the salt and cornmeal. Reduce the heat to low and cook, whisking constantly, until the polenta is smooth and slightly thickened, about 2 minutes.

Step 7

Cover and cook, stirring every 10 minutes and scraping up the sides, bottom, and corners of the pan, until creamy and tender, about 30 minutes total. Remove from the heat, stir in the Parmesan, and cover to keep warm.

Step 8

To serve, divide the polenta between bowls. Top with a lamb shank, then spoon the sauce over the lamb and polenta.