Preheat the oven to 325 degrees F and spread the rye and cornbread cubes out on 2 rimmed baking sheets. Bake until golden and toasted, 20 to 25 minutes. Let cool. Lower the oven to 375 degrees F.
Cook the sausage meat in a skillet over medium-high heat, breaking up with a spoon, until browned, 5 minutes. Set aside.
Grease a 9- by 13-inch baking dish with butter or gluten-free cooking spray. In a large skillet, melt the butter over medium heat. Add the onion, celery and mushrooms; season with salt and pepper, and cook until softened and the liquid has evaporated, about 10 minutes. Transfer to a large bowl. Add the toasted rye cubes, toasted cornbread cubes, cooked sausage and about 3/4 teaspoon salt and 1/4 teaspoon pepper.
In a medium bowl, whisk together the broth, cream, eggs, raisins, parsley and caraway seeds; pour over the bread mixture and toss to evenly coat. Transfer to the prepared baking dish and bake, uncovered, until golden, about 30 minutes. Serve immediately.