01 - Preheat the 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 the sliced 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. Mix in the grated carrots, salt, black pepper, caraway seeds, and smoked paprika. Stir to combine evenly.
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. Top with half of the sauerkraut mixture. Repeat with the remaining cabbage and sauerkraut to form a second layer.
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. Place in the oven and slow roast for 1 hour.
08 - Remove the foil and increase the oven temperature to 375°F. Continue baking 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, allowing the cheese to melt.
10 - Remove from the oven and let the casserole rest for 10 minutes before serving. Garnish with additional fresh dill.