1. Bring butter, salt, pepper, and 1 cup water to a boil over high heat in a deep 12-inch heavy skillet, then add Brussels sprouts and simmer, partially covered, stirring occasionally, until crisp-tender, 6 to 8 minutes.
2. Remove lid and boil over moderately high heat until water is evaporated and sprouts are lightly browned, 3 to 4 minutes.
3. Add cream and remaining 1/4 cup water and bring to a boil, stirring.
4. Add chestnuts, then reduce heat and simmer, stirring occasionally, until heated through, about 2 minutes.

Serves 8.

1. Heat oven to 400 degrees. Using a paring knife or a chestnut knife, cut a large X into the shell of each chestnut on one side. Arrange on a baking pan in a single layer, cut side up. Roast until flesh is tender, 20 to 25 minutes. Remove from oven. Immediately remove and discard shells, keeping chestnuts whole if possible. Set chestnuts aside.
2. Melt butter and oil in a large saute pan set over medium-high heat. Add brussels sprouts; season with salt and pepper. Cook, stirring occasionally, until golden, 16 to 18 minutes.
3. Add roasted chestnuts. Cook, gently stirring occasionally, until brussels sprouts are tender and spotted deep brown, 20 to 25 minutes.
4. Add vinegar, sugar, and turkey stock. Cook, stirring occasionally, until liquid has been reduced to a syrup, 4 to 5 minutes. Transfer to a serving dish, and serve.

Makes 8 servings.