Crumble cooled cornbread into a large bowl using your hands
Add crumbled white bread or biscuits if using and mix gently
Melt butter in a skillet over medium heat
Add onion and celery and cook 6–8 minutes until soft and fragrant
Let vegetables cool slightly, then add to the cornbread mixture
Add salt, black pepper, sage, poultry seasoning, and thyme if using
Stir gently to distribute seasoning evenly
Add beaten eggs and mix lightly
Slowly pour in chicken broth, starting with 2½ cups, stirring gently
Stop adding broth when mixture is very moist but not soupy
Preheat oven to 375°F (190°C)
Butter a 9×13-inch baking dish
Spoon dressing into dish and spread evenly
Bake uncovered for 40–45 minutes until set and lightly golden
Cover loosely with foil if top browns too quickly