Preheat the oven to 400 degrees F.
In a large sauté pan, over medium heat, add 2 tablespoons of the oil. When the oil is hot add the eggplant. Season with salt and cayenne. Sauté for 3 to 4 minutes, or until slightly wilted. Add the onions and celery. Season with salt and cayenne. Continue to sauté for 3 minutes. Remove from the heat and cool completely. In a mixing bowl, combine the eggplant mixture, garlic and cornbread. Mix well. Stir in enough of the chicken stock to make a moist but not overly wet dressing.
Rub the remaining tablespoon of oil on both sides of each fillet and season lightly with salt and cayenne. Place the fish, skin side up, on a parchment paper lined sheet pan. Place 1 cup of the dressing on half of each fillet. Fold the fillets in half over the dressing and press down lightly to secure. Place the fish in the oven and bake until the fish is opaque and the dressing is golden brown, about 15 minutes. Remove from the oven and transfer to plates. Garnish with parsley and serve with the sauce piquante.