Heat the oven to 350° and coat a 9-by 13-inch baking dish with butter. In a medium bowl, whisk together the first six ingredients.
With a hand mixer or a stand mixer with paddle attachment, beat the butter until smooth. Add the dry ingredients, and blend at medium speed until the mixture resembles coarse crumbs, about 1 minute. Reduce the mixer's speed to low and add the eggs, one at a time, until just blended. Slowly add the buttermilk and blend evenly.
Pour the batter into the pan, and bake until the top is golden brown and a toothpick inserted in the middle comes out clean, about 20 minutes. Let bread cool for 15 minutes.
Break the cooled cornbread into pieces, and set aside. Heat the oven to 375° and coat a 9-by 13-inch baking dish with butter.
In a large pan over medium heat, warm the olive oil. Add the sausage and cook, breaking it into pieces with a spoon, until browned, about 8 minutes. Transfer it to a paper towel-lined plate to drain.
In the same pan, melt the butter. Add the onions and celery, and cook until soft, about 5 minutes. Add the apples, and cook until slightly soft, about 3 minutes. Add the sage and thyme, season with salt and pepper, and cook for 1 minute more.
In a large bowl, combine the cooked apple-herb mixture, sausage, cornbread, and parsley. Gently stir to combine. Add the chicken broth and eggs, and toss lightly to coat. Transfer the mixture to the prepared dish, spread evenly, and bake until golden brown on top, about 30 minutes. Let dressing cool for 15 minutes before serving.