01 - Preheat the oven to 325°F. Grease a large 3-quart casserole dish with a light coating of olive oil.
02 - In a large skillet over medium heat, warm the olive oil. 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. Fold in the grated carrots, salt, black pepper, caraway seeds, and smoked paprika. Toss to combine evenly.
04 - In a mixing bowl, combine the drained sauerkraut with the grated apple and chopped dill. Stir until evenly incorporated.
05 - Spread half of the cabbage mixture across the bottom of the prepared casserole dish. Layer half of the sauerkraut mixture over the cabbage. Repeat with the remaining cabbage mixture followed by the remaining sauerkraut mixture.
06 - Pour the 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 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 sour cream and cheese, spread the sour cream evenly over the top and sprinkle with the 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.