Preheat the oven to 350 degrees F.
Cut the mirlitons in half and scoop out the seeds.
Bring a large pot of salted water to a boil and add the mirliton halves. Cook for 10-15 minutes, or until they are tender.
In a large skillet, melt the butter over medium heat. Add the onion, green bell pepper, and celery, and cook until they are soft.
Stir in the flour, salt, cayenne pepper, and black pepper. Gradually add the milk and chicken broth, stirring constantly, until the mixture thickens.
Stir in the shrimp and 1/2 cup of Parmesan cheese.
Stuff the mirlitons with the shrimp mixture and place them in a baking dish.
Sprinkle the remaining Parmesan cheese on top of the mirlitons.
Bake for 20-25 minutes, or until the cheese is melted and the mirlitons are heated through.