01 - Preheat oven to 325°F. Grease a large casserole dish with 1 tablespoon of the olive oil, ensuring even coverage across the bottom and sides.
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. Stir in the minced garlic and grated carrots, then cook for an additional 2 minutes until fragrant.
03 - Arrange half of the sliced cabbage in an even layer across the bottom of the prepared casserole dish. Spread half of the drained sauerkraut over the cabbage, followed by half of the sautéed vegetable mixture. Season with salt, pepper, and 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 deeply tender.
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 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 at 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 parsley if desired.