Heat oil in a large pot or Dutch oven over medium heat. Add onions and cook, stirring often, until light golden, 6 to 8 minutes.

Add cabbage and cook, stirring occasionally, until the cabbage wilts, about 5 minutes. Stir in broth, sugar and caraway seeds; bring to a simmer. Reduce heat to low, cover and cook until the cabbage is very tender, 15 to 20 minutes.

Stir in chestnuts, vinegar, salt and pepper. Increase heat to medium and cook, uncovered, until most of the liquid has evaporated and the chestnuts are hot, 5 to 8 minutes.

Ingredient Note: You can find convenient cooked chestnuts in cans or vacuum-sealed containers at specialty stores or through mail-order; try amazon.com (Gourmet Food tab), chestnutsonline.com.

To cook chestnuts: Eight ounces of fresh chestnuts yields about 1 cup peeled and cooked. Using a small knife, score a cross on the flat side of each chestnut. Using a slotted spoon, dip chestnuts, 4 or 5 at a time, into a saucepan of boiling water. Peel away shells and inner brown skins. If chestnuts are difficult to peel, return them to boiling water for a few seconds. Place peeled chestnuts in a large saucepan and add enough water to cover. Simmer, covered, until tender, 30 to 45 minutes. Drain and refresh with cold water.

MAKE AHEAD TIP: Refrigerate cabbage for up to 2 days. Reheat in a casserole with 1/2 cup water, covered, at 350 degrees F for 40 to 50 minutes.