01 - Preheat the oven to 325°F. Grease a 3-quart casserole dish with a light coating of olive oil.
02 - Heat olive oil in a large skillet over medium heat. Sauté the onions until translucent, about 5 minutes. Add the minced garlic and cook for 1 additional minute until fragrant.
03 - Add the shredded cabbage to the skillet and cook, stirring occasionally, until slightly wilted, about 5 to 7 minutes. Stir in the grated carrots, salt, pepper, caraway seeds, and smoked paprika until evenly combined.
04 - In a mixing bowl, combine the drained sauerkraut with the grated apple and chopped dill. Toss until well blended.
05 - Spread half of the cabbage mixture evenly across the bottom of the prepared casserole dish. Layer half of the sauerkraut mixture over the cabbage. Repeat with the remaining cabbage mixture followed by the remaining sauerkraut mixture.
06 - Pour the vegetable broth and white wine, if using, evenly over the assembled casserole layers.
07 - Cover the dish tightly with aluminum foil or a lid. Place in the oven and slow roast for 1 hour.
08 - Remove the foil and increase the oven temperature to 375°F. Bake uncovered for 30 to 45 minutes until the top is lightly browned and most of the liquid has evaporated.
09 - If using sour cream and cheese, spread the sour cream over the top and sprinkle with grated cheese during the final 10 minutes of baking.
10 - Remove from the oven and let rest for 10 minutes before serving. Garnish with fresh dill.