Cut the cornbread into 1-inch pieces and place on a large baking sheet. Cut the white bread into 1-inch cubes and add to a separate baking sheet.
Option 1 - Preheat oven to 250° F. Place the pans in the warm oven and bake until dry, about 45 - 50 minutes. Rotate and switch the pans halfway through baking.
Option 2 - Instead of oven-drying, stale the cubed breads overnight at room temperature.
Melt 6 tablespoons butter in a large nonstick skillet over medium-high heat. Add the onions and celery and cook until softened, about 5 minutes. Add the sage and thyme and sauté until fragrant, about 1 minute. Remove from the heat and cool slightly.
In a large mixing bowl whisk together the chicken broth, eggs, salt and pepper. Add the sautéed onion and celery mixture to the chicken broth. Stir until combined.
Add the dried cornbread and bread cubes to the broth. Gently fold together until combined. Allow the mixture to rest for 5 - 10 minutes, gently stirring a few times, or until most of the liquid is absorbed.
Preheat oven to 375° F. Lightly grease a 13x9-inch baking pan. Gently scoop the dressing into the prepared pan. Drizzle 2 tablespoons of melted butter on top and bake for 40-45 minutes. Cool 15 minutes before serving.