Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch casserole dish. Spread French bread cubes onto a baking sheet.
Bake bread cubes in the preheated oven until bread is toasted, 10 to 15 minutes.
Heat a large skillet over medium-high heat. Cook and stir sausage in the hot skillet until browned and crumbly, 5 to 7 minutes. Transfer sausage to a bowl using a slotted spoon; reserve drippings in the skillet.
Cook and stir celery, onion, and garlic in the skillet with sausage drippings over medium heat until vegetables are soft, 5 to 10 minutes.
Put crumbled corn bread in a large bowl; add bread cubes, celery mixture, sausage, cream of chicken soup, shredded chicken, 1 sleeve crushed buttery round crackers, and eggs. Stir well. Pour in enough chicken broth to make mixture extremely moist. Spoon mixture into the prepared casserole dish. Sprinkle 1/4 cup crushed buttery round crackers over casserole; drizzle with melted butter.
Bake in the preheated oven until cooked through and topping is lightly browned, about 45 minutes.