Preheat oven to 350°. Line a rimmed baking sheet with parchment paper.
Place cornbread on prepared pan. Toast until lightly browned and crisp, 15 to 20 minutes. Set aside.
In a 12-inch enamel-coated cast-iron braiser, heat butter over medium-high heat. Add andouille; cook until lightly browned, 3 to 4 minutes. Remove sausage using a slotted spoon, and let drain on paper towels.
Add onion, celery, bell pepper, and garlic to braiser; sauté until vegetables are softened, 7 to 9 minutes. Stir in giardiniera, Creole seasoning, thyme, sage, and pepper.
In a large bowl, whisk together broth and eggs. Add cornbread, andouille, and vegetable mixture, stirring gently until liquid is absorbed. Spoon mixture back into braiser, and drizzle with melted butter.
Bake until golden brown and firm to the touch, 25 to 30 minutes. Serve immediately.