01 - Preheat oven to 325°F. Lightly grease a 3-quart casserole dish with olive oil.
02 - Heat olive oil in a large skillet over medium heat. Sauté onions until translucent, about 5 minutes. Add minced garlic and cook for 1 additional minute until fragrant.
03 - Add sliced cabbage to the skillet and cook, stirring occasionally, until slightly wilted, about 5 to 7 minutes. Stir in grated carrots, salt, pepper, caraway seeds, and smoked paprika until evenly combined.
04 - In a mixing bowl, combine drained sauerkraut with 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 remaining cabbage and sauerkraut to form a second layer.
06 - Pour vegetable broth and white wine evenly over the assembled casserole. Cover tightly with aluminum foil or a lid and slow roast in the oven for 1 hour.
07 - Remove the foil, increase 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.
08 - If using, spread sour cream evenly over the top and sprinkle with grated cheese during the final 10 minutes of baking, allowing the cheese to melt and bubble.
09 - Remove from oven and let rest for 10 minutes before serving. Garnish with fresh chopped dill.