01 - Preheat the oven to 325°F. Grease a large casserole dish with 1 tablespoon of olive oil and set aside.
02 - Heat the remaining 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, and continue cooking for 2 more minutes until fragrant. Remove from heat.
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 onion-carrot 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, allowing it to seep down between the vegetables.
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 together.
06 - While the casserole roasts, combine the breadcrumbs with the melted butter in a small bowl, tossing until the crumbs are evenly coated and resemble wet sand.
07 - Remove the casserole from the oven and carefully take off the foil. Sprinkle the buttered breadcrumbs in an even layer across the top. Return the uncovered casserole to the oven 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 rest for 10 minutes before serving to allow the layers to set. Garnish with fresh chopped parsley if desired, and serve warm.