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