Preheat oven to 350 F.
Cut corn bread into rough, 1 inch chunks. Combine with white bread chunks and toss to mix.
Melt butter in a skillet over medium heat. Add onions, celery, and 1/2 tsp salt. Cook until transparent, about 5-10 minutes.
Transfer vegetables to bread mixture and combine.
Beat the eggs then add to the bread mix. Sprinkle in sage and thyme, add salt and pepper to taste. Toss together.
Add 4 cups broth and stir well. Using your hands, work the mixture into a batter-like consistency. Don't be afraid of breaking up the bread. Add another cup if the mixture is dry or not pourable.
Lightly grease a baking dish (anything that holds 2 quarts. I like 9x13, 8x11 works too). Transfer mixture to the greased dish and bake until the dressing has browned around the edges, about 45 minutes to 1 hour.