1 Cook the bacon: Cook the bacon batons slowly in a large sauté pan (with cover) on medium low heat until most of the fat has rendered out, and the bacon is lightly browned, not crisp.
2 Add onions: Add the sliced onions, increase the heat of the pan to medium high and cook, stirring often, until the onions have softened and lightly browned.
3 Add cabbage, apples, vinegars, sugar: While the onions are cooking, peel the apple and cut it into 1/2-inch thick pieces. Once the onions are ready, add the sliced cabbage, chopped apple, cider vinegar, balsamic vinegar, and sugar.
Carefully toss to combine. Bring to a simmer, cover the pan, lower the heat, and simmer until the cabbage is cooked through and soft, about 40 minutes.
4 Add chestnuts: Uncover the pan. Add the chopped roasted chestnuts. Add salt and pepper to taste. Increase the heat to boil off excess moisture for a few minutes.