Preheat the oven to 350 degrees F. Rub the mirlitons with oil. Place them on a baking sheet cut side down and bake until they are fork tender and easily peeled, about 45 minutes. Set the mirlitons aside to let rest until they are cool enough to handle, then peel and cut them into 1-inch pieces.
Melt the butter in a large skillet over moderate heat. Add the onions, celery, and bell peppers and cook until the onions are translucent, about 5 minutes. Add the garlic and cook for another minute. Increase heat to medium-high, add the fresh herbs and shrimp, and stir frequently, until shrimp are just cooked , 3-5 minutes. Stir in the crabmeat. Transfer to a large mixing bowl. Add the diced mirlitons and the remaining ingredients and stir until well combined.
Spoon the dressing into a large buttered baking dish and bake until golden brown, 20-30 minutes.