Preheat the oven to 300°F. Line a large, rimmed baking sheet with parchment paper.
Cut the day-old cornbread into 1-inch cubes (you will have about 9 cups).
Arrange the cubed cornbread on the prepared baking sheet. Add the cubed sandwich bread to the pan. Bake until lightly toasted, about 10 minutes. Set aside to cool.
In a very large bowl, whisk together the broth and eggs. Set aside.
In a large skillet, melt the butter over medium heat. Add the onion, celery, and carrots; sauté until soft and fragrant, about 10 minutes. Remove from the heat and let cool slightly.
Stir the vegetables (and any liquid from the skillet) into the broth and egg mixture. Add the parsley, poultry seasoning, sage, thyme, salt, and pepper. Gently fold in the cornbread and bread cubes until all of the ingredients are well combined. The mixture should be moist but not soupy; add extra broth if needed.
Increase the oven temperature to 350°F. Transfer the mixture to a greased 9 x 13-inch baking dish. Cover with foil and bake for 20 minutes.
Remove the cover and continue baking until the dressing is set and the top is toasted and golden brown, about 25-30 more minutes. Garnish with chopped fresh herbs just before serving.