While the cornbread is toasting, heat a large skillet over medium-high heat and add 1 tablespoon of the butter. Once the butter has melted, add the sausage and cook until browned, for about 7-10 minutes, breaking it up into smaller chunks using a wooden spoon or spatula. Transfer the sausage to a paper towel-lined plate and set aside.
Increase the oven temperature to 350°F and liberally butter a 9×13-inch baking dish with 2 tablespoons of butter.
In the same skillet, add 1 stick of butter over medium-high heat. Once melted, add the onion and sauté for 5-7 minutes, until starting to become golden brown (I like to give it a headstart so it gets a bit more brown and has more flavor). Season with salt, pepper, celery leaves, sage, and thyme. Then add the celery and continue sautéing with the onion, stirring occasionally for about 5 more minutes. Add the white wine and then cook for about 5 minutes more to let the wine reduce. Be sure to scrape up any brown bits on the bottom of the pan.
In a large bowl, whisk the eggs well, then whisk in the chicken stock. Add the parsley and cornbread cubes and toss very gently, using a spatula, taking care not to let the cornbread crumble.
Transfer the cornbread dressing to the prepared baking dish. Cover with foil (or a lid) and place in the oven to bake for 25 minutes. Then remove the foil and continue baking for about 15 minutes more, or until the top is golden brown and the internal temperature of a thermometer reads 160°F.