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 a sprinkle of caraway seeds if using. Repeat the layering process 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 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 breadcrumbs are evenly coated.
07 - Remove the foil from the casserole. Sprinkle the buttered breadcrumb mixture 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 from the oven and let the casserole stand for 10 minutes before serving to allow it to set. Garnish with fresh parsley if desired.