1. Preheat oven to 350 degrees. Using a chestnut knife or a small paring knife, make an incision about 1/8 inch deep through the shell and into the flesh of each chestnut almost all the way around the circumference of the nut.Transfer to a chestnut pan or rimmed baking pan. Roast in the oven until the chestnuts are tender, about 35 minutes.Turn the oven off. Leaving the pan with the chestnuts in the oven, remove several at a time.Working quickly, place 1 chestnut in a towel, and, holding both, peel the chestnut while still hot. Remove and discard the shells and inner skin; set aside.

2. Season the lamb with salt and pepper. Dredge the meat in flour, and shake off excess; set aside. Heat 2 tablespoons oil in a Dutch oven over medium-high heat. Add lamb, in a single layer without crowding, and cook until browned all over, about 5 minutes.Transfer lamb to a dish. Repeat browning with the remaining olive oil and meat.

3. To the Dutch oven, add onions, and cook until golden, about 4 minutes. Add mushrooms, season with salt and pepper, and cook until well browned, 5 to 6 minutes. Add red wine, scraping any browned bits from the bottom of the pan. Cook until the wine is reduced by half, about 5 minutes.

4. Add the reserved lamb, beef stock, chestnuts, tomatoes, sage, and rosemary, and bring to a boil. Reduce heat to medium-low, and simmer, uncovered, until the meat is very tender, 1 1/2 to 2 hours. Add cranberries, and cook 2 minutes. (Proceed with step 5, and serve immediately, or let cool, refrigerate overnight, and proceed with step 5 just before serving.)

5. Serve the hot polenta with the lamb-and-chestnut stew.