01 - Preheat the oven to 325°F. Grease a large casserole dish with 1 tablespoon of olive oil, coating the bottom and sides evenly.
02 - In a large skillet, heat the remaining olive oil over medium heat. Add the sliced onion and cook, stirring occasionally, until softened and translucent, 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. Layer half of the sauerkraut over the cabbage, followed by half of the sautéed vegetable mixture. Season with salt, pepper, and a sprinkle of caraway seeds if using. Repeat the layers with the remaining cabbage, sauerkraut, and sautéed vegetables, seasoning the top layer as well.
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. Sprinkle the buttered breadcrumbs evenly over the top. Return to the oven and bake, uncovered, for an additional 30 minutes, or until the topping is golden brown and the casserole is bubbling at the edges.
08 - Remove the casserole from the oven and let it rest for 10 minutes before serving to allow the layers to set. Garnish with fresh parsley if desired.