01 - Preheat oven to 325°F. Grease a large casserole dish with 1 tablespoon of the olive oil, coating the bottom and sides evenly.
02 - Heat the remaining 2 tablespoons of olive oil in a large skillet over medium heat. Add the sliced onion and cook, stirring occasionally, until softened and translucent, about 5 minutes. Stir in the minced garlic and grated carrots, continuing to cook for 2 more minutes until fragrant.
03 - Arrange half of the cabbage wedges in the prepared casserole dish. Spread half of the sauerkraut over the cabbage, then distribute half of the sautéed onion-carrot mixture on top. Season with salt, pepper, and caraway seeds if using. Repeat with the remaining cabbage, sauerkraut, and sautéed vegetables to form a second layer.
04 - In a mixing bowl, whisk together the sour cream, eggs, and vegetable broth until smooth and well combined. Pour the mixture evenly over the assembled casserole layers.
05 - Cover the casserole dish tightly with aluminum foil. Place in the oven and roast for 1 hour and 30 minutes, allowing the cabbage to become tender and the flavors to meld.
06 - While the casserole roasts, combine the breadcrumbs with the melted butter in a small bowl, tossing until the crumbs are evenly coated.
07 - Remove the foil from the casserole and sprinkle the buttered breadcrumbs evenly over the top. Return to the oven uncovered and bake for an additional 30 minutes, or until the topping is golden brown and the casserole is bubbling around the edges.
08 - Remove from the oven and let the casserole stand for 10 minutes before serving to allow the layers to set. Garnish with fresh chopped parsley if desired.