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 shredded cabbage to the skillet and cook, stirring occasionally, until slightly wilted, about 5 to 7 minutes. Mix in grated carrots, salt, black 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 evenly incorporated.
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 and sauerkraut to form a second layer.
06 - Pour 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 and increase 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, spread sour cream evenly over the top and sprinkle with grated cheese during the last 10 minutes of baking.
10 - Remove from oven and let rest for 10 minutes before serving. Garnish with additional fresh dill.