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 sliced onions until translucent, approximately 5 minutes. Add minced garlic and cook for 1 additional minute until fragrant.
03 - Add shredded cabbage to the skillet, stirring occasionally until slightly wilted, about 5 to 7 minutes. Fold in grated carrots, salt, black pepper, caraway seeds, and smoked paprika. Mix thoroughly to combine.
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 remaining cabbage and sauerkraut to form a second layer.
06 - Pour vegetable broth and white wine, if using, evenly over the assembled casserole layers.
07 - Cover the casserole dish tightly with aluminum foil or a fitted lid. Roast in the oven at 325°F for 1 hour.
08 - Remove the foil and increase 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 sour cream and cheese, spread sour cream evenly over the top and sprinkle with grated cheese during the final 10 minutes of baking.
10 - Remove from oven and let rest for 10 minutes before serving. Garnish with fresh chopped dill.