Heat the oven to 350°F and grease a 9-by-13-inch baking dish with 1 tablespoon butter. Combine the crumbled cornbread and white bread in a large bowl.
Melt the remaining butter in a skillet over medium heat. Cook onion and celery for 8 to 10 minutes until soft, then stir in sage, thyme, salt and pepper.
Fold the vegetables into the bread. Whisk broth and eggs together and fold into the mixture; let stand for 10 minutes so the crumbs absorb the liquid.
Spread in the dish and cover with foil. Bake for 25 minutes, then remove the foil.
Bake for 15 to 20 minutes more until golden at the edges and 165°F in the center. Rest for 10 minutes before serving in soft spoonfuls.