In a large Dutch oven, melt 1 tablespoon butter over medium heat. Add 1/3 pound chopped thick-cut bacon and cook until crisp, 5 to 7 minutes; using a slotted spoon, transfer to paper towels to drain. Add 1 chopped onion and 2 chopped carrots to the pot and cook until softened, 3 to 5 minutes. Stir in 2 chopped granny smith apples and 2 teaspoons sugar; cook for 1 minute. Stir in 5 cups shredded cabbage (about 1/2 head) and cook for 2 minutes more. Increase the heat to high and stir in 1/4 cup white balsamic vinegar or white wine vinegar and 1/3 cup apple cider; cook until the liquid is reduced by half, 3 to 4 minutes. Add 1/2 cup chicken broth and bring to a boil. Cover and bake in a 350 degrees oven until the cabbage is silky, about 1 hour. Remove the lid and continue cooking until the cabbage is browned around the edges, 5 to 10 minutes. Stir in the reserved bacon.