01 - Preheat the oven to 325°F. Grease a 3-quart casserole dish lightly with 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, black 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 and finish with the remaining sauerkraut mixture on top.
06 - Pour the vegetable broth and white wine, if using, evenly over the assembled casserole.
07 - Cover the dish tightly with aluminum foil or a lid. Roast in the oven for 1 hour at 325°F.
08 - Remove the foil, increase the oven temperature to 375°F, and bake uncovered for 30 to 45 minutes until the top is lightly browned and most of the liquid has evaporated.
09 - If using, spread the sour cream evenly over the top and sprinkle with grated cheese during the last 10 minutes of baking.
10 - Remove from the oven and let the casserole rest for 10 minutes before serving. Garnish with additional fresh dill.