01 - Preheat the oven to 325°F (165°C). Grease a large casserole dish with 1 tablespoon of olive oil.
02 - In a large skillet, heat the remaining olive oil over medium heat. Add the sliced onion and cook, stirring occasionally, until softened, about 5 minutes. Add the minced garlic and grated carrots; continue cooking for 2 more minutes until fragrant.
03 - Arrange half of the sliced cabbage in the prepared casserole dish. Top with half of the sauerkraut, followed by half of the sautéed vegetable mixture. Season with salt, pepper, and caraway seeds if using. Repeat the layers with the remaining cabbage, sauerkraut, and sautéed vegetables.
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 layered casserole.
05 - Cover the casserole dish tightly with aluminum foil. Place in the oven and roast for 1 hour and 30 minutes, allowing the flavors to meld and the cabbage to become tender.
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.
08 - Remove the casserole from the oven and let it stand for 10 minutes before serving to allow the layers to set. Garnish with fresh parsley if desired.