Preparation
Pre-heat the oven to 375. Lightly spray a 9-inch pie plate or baking dish with vegetable oil spray.
Pour 1 tablespoon olive oil into a 4-quart saucepan or Dutch oven and swirl to coat the bottom. Heat over medium-high heat. Cook the onion and garlic for 2 to 3 minutes, or until they begin to soften. Stir in the celery and carrots. Cook for 5 minutes, or until the celery begins to soften. Transfer the vegetables to a plate.
In the same saucepan, heat 2 tablespoons olive oil over medium heat. Whisk in the flour, thoroughly incorporating the oil. (The mixture will be dry.) Gradually pour in 1 cup broth, whisking constantly. Cook for 2 to 3 minutes, or until the mixture begins to thicken and turn golden and the flour loses its raw taste.
Stir in the chicken, peas, parsley, and enough of the remaining broth to achieve the desired consistency. Stir in the salt and pepper. Pour the mixture into the pie pan.
Spread the mashed potatoes over the chicken mixture.
Bake for 25 minutes, or until the potato crust is golden and the filling is bubbly. Allow to rest for at least 5 minutes before serving.