Prepare the cornbread:
Heat oven to 375°. Grease an 8-inch baking dish with cooking spray or butter and set aside. In a medium bowl, whisk the cornmeal, flour, sugar, baking powder and salt. In a large bowl, whisk the melted butter, buttermilk and egg until combined. Slowly add the dry ingredients to the wet mixture and stir with a wooden spoon until the thick batter is fully mixed. Spread the batter in the prepared dish, smooth the top and bake until lightly golden and a toothpick inserted in the center comes out clean, about 25 to 30 minutes. Let the cornbread cool completely, then cut it into 1- to 1 1/2-inch cubes. The cornbread can be made a day ahead and left to sit overnight.

Prepare the dressing:
Grease a 9-inch square baking dish with butter and set aside. In a skillet over medium heat, heat the olive oil and cook the sausage, breaking it up with a wooden spoon, until golden and cooked through, about 8 minutes. Transfer the sausage to a large mixing bowl and pour off all but 2 Tbsp. of the rendered fat. Add the butter, onion and celery to the skillet and cook until slightly softened, about 5 minutes. Stir in the thyme and sage, then season with salt and pepper. Slowly add the broth to deglaze the pan, scraping up the browned bits. Remove from heat, let cool slightly and add to the mixing bowl with the sausage. Whisk the eggs in a small bowl. Add the cornbread cubes to the meat and vegetable mixture and stir gently, then pour in the eggs and stir until incorporated. Add a little more broth if the mixture seems too dry. Taste the mixture and adjust seasoning. Transfer to the prepared baking dish and bake until golden, crisp on top and set in the center, about 30 minutes. Garnish with parsley before serving.