For the Cornbread

1. Preheat the oven to 350°F and set an oven rack in the middle position. Butter and flour a 9x13-inch baking pan or spray with a nonstick cooking spray with flour, such as Baker's Joy.
2. In a medium bowl, whisk the eggs. Add the milk and whisk until evenly combined. Set aside.
3. In a large bowl, whisk together the flour, cornmeal, sugar, baking powder, and salt. Add the milk mixture and the melted butter to the dry ingredients and whisk until smooth. Transfer the batter to the prepared baking pan and smooth the top. Bake until lightly golden around the edges and set, about 30 minutes. Cool in the pan on a wire rack until cool enough to handle, 20 to 30 minutes.
4. Preheat the oven to 250°F and set the oven rack in the middle position. Cut the cornbread into 1-inch squares. Place the cut cornbread on a rimmed baking sheet and bake until lightly toasted, about 1 hour. Cool the cornbread on the baking sheet, about 15 minutes.

For the Stuffing

1. Place the sausage in a large nonstick pan over medium-high heat. Cook, stirring and breaking it apart with a wooden spoon, until no longer pink, about 5 minutes. Using a slotted spoon, transfer the sausage to the largest bowl you have (it needs to fit all the cornbread), leaving the fat in the pan. Reduce the heat to medium and add the onions, celery, and 3 tablespoons of the butter to the pan. Cook until the vegetables are softened, 6 to 8 minutes. Add the garlic and cook 1 minute more. Off the heat, mix in the herbs. Add the vegetable mixture to the bowl with the sausage and mix to combine.
2. In another bowl, whisk the eggs. Add the milk, broth, and salt.
3. Preheat the oven to 350°F and set an oven rack in the middle position. Grease a 9x13-inch baking pan with butter or nonstick cooking spray.
4. Add the dried cornbread cubes to the sausage and vegetable mixture. Pour the egg mixture over top. Fold the mixture a few times with a large rubber spatula, being careful not to break up the cornbread too much. Let the mixture sit, folding carefully a few times, until the cornbread is saturated, about 10 minutes.
5. Transfer the soaked cornbread mixture to the prepared pan, making sure the sausage and vegetables are evenly distributed. Try to arrange some of the larger pieces of cornbread on top; it looks prettier. Dot the top of the stuffing with the remaining 3 tablespoons of butter. Bake until golden brown and crisp, about 35 minutes. Serve warm.
6. Make-Ahead/Freezer-Friendly Instructions: The cornbread can be prepared up to two days ahead of time. Cut it into cubes and let it dry out on the countertop overnight instead of toasting it in the oven. If you'd like the assemble the stuffing in advance, prepare it up until the point of baking, cover it with plastic wrap, and refrigerate up to a day ahead of time. When ready to serve, bake as directed, allowing a few extra minutes in the oven. It can also be frozen after baking, tightly covered, for up to 3 months. When you’re ready to serve it, defrost in the refrigerator for 24 hours. Reheat it, covered with foil, in a 325°F oven until hot.