01 - Preheat the oven to 325°F. Grease a large casserole dish with 1 tablespoon of the olive oil, coating the bottom and sides evenly.
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.
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 completely smooth and well combined. Pour the mixture evenly over the assembled casserole layers.
05 - Cover the casserole dish tightly with aluminum foil and place in the preheated oven. Roast for 1 hour and 30 minutes until the cabbage is fork-tender.
06 - While the casserole roasts, combine the breadcrumbs and melted butter in a small bowl, tossing until the crumbs are evenly coated and resemble wet sand.
07 - Remove the foil from the casserole and sprinkle the buttered breadcrumbs in an even layer over the top. Return to the oven uncovered and bake for an additional 30 minutes, or until the topping is deep golden brown and the casserole is bubbling at the edges.
08 - Remove the casserole from the oven and let it rest for 10 minutes to allow the layers to set before serving. Garnish with freshly chopped parsley if desired.