Heat the oil in a 10-inch skillet over medium heat. Add the onion, mushrooms and celery and cook until the vegetables are tender. Remove the skillet from the heat.
Heat the butter in a 10-inch oven-safe skillet over medium heat. Stir in the flour. Gradually stir in the half-and-half and stock. Cook and stir until the mixture boils and thickens. Stir in the sherry, thyme, parsley and garlic powder. Season the sauce with the cayenne pepper, if desired.
Stir the turkey, chicken, onion mixture, carrots, peas, vermicelli and Parmesan in the skillet with the sauce.
Broil 4 inches from the heat until the mixture is hot and bubbling and the top is golden.