Break cornbread into coarse crumbs onto 2 rimmed baking sheets. Set out uncovered overnight to dry.
Preheat oven to 350°. Grease a 9- by 13-in. baking dish. Put cornbread in a large bowl.
Melt butter in a large frying pan over medium heat. Add celery, onions, thyme, and sage; cook, stirring occasionally, until vegetables are tender, about 15 minutes. Stir in salt and pepper.
Add vegetables to cornbread; stir in enough broth until cornbread is moistened but not soggy. Stir in eggs. Put stuffing in baking dish; cover with foil.
Bake stuffing 30 minutes. Uncover and bake until crisp on top, 25 to 30 minutes more.
*Follow recipe for Easy Chicken Broth at sunset.com, with chicken or turkey bones.
Make ahead: Through step 1, up to 2 days (transfer dry crumbs to a container), or through step 4, up to 1 day, chilled; bake, covered (step 5), 45 minutes before uncovering.