Preheat your oven to 375°F. Butter a 9 x 13 baking dish and set it aside.
In a 10-inch nonstick or cast-iron skillet, brown the smoked sausage in 1 tablespoon of olive oil over medium heat and set aside.
Add 2 tablespoons of butter, onions, bell pepper, and celery to the same skillet and cook for 2 to 3 minutes until softened. Add the crawfish, garlic, and 2 teaspoons of Cajun seasoning and cook for 2 to 3 more minutes until crawfish are heated.
Transfer the cornbread, crawfish mixture, browned sausage, remaining Cajun seasoning, poultry seasoning, salt and pepper, and 3 cups of chicken broth to a large mixing bowl. Gently mix and add more broth until desired texture is reached. The dressing should have the texture of thick oatmeal, wet but not soupy. Stir in beaten eggs.
Transfer the dressing to the buttered baking dish, smooth into an even layer, and top with the remaining 2 tablespoons of butter, sliced into thin slivers.
Bake the dressing, covered, for 35 minutes then uncovered for 15 to 20 minutes until the center is no longer giggly and the edges are browned. Garnish with fresh chopped parsley.