Preheat oven to 400° Fahrenheit. Spray 9x9 baking dish with cooking spray. Set aside.
Melt 4 tablespoons butter in large skillet. Sauté 2 small chopped potatoes, 1 stalk diced celery, 1 chopped carrot, ½ cup frozen peas until crisp-tender, approximately 5 minutes.
Sprinkle 4 tablespoons all purpose flour, salt, freshly cracked black pepper, 1 teaspoon garlic powder, 1 tablespoon dried thyme, pinch nutmeg. Stir to coat all vegetables.
Cook one minute, stirring constantly. Gradually add 2 cups chicken broth, 1 cup milk; cook over medium heat, stirring constantly until thick. Add 2 cups chopped roasted chicken. Taste for seasoning, adding additional salt or freshly cracked black pepper as desired. Pour filling into prepared baking dish.
Unfold thawed puff pastry onto lightly floured surface. Gently press together seams, rolling out to extend size to fully cover filling. Place puff pastry over filling.
Add 1 egg and 1 teaspoon water to small bowl. Whisk together. Use pastry brush to brush egg mixture over top of puff pastry. Use sharp knife to cut several slits into puff pastry.
Bake 30 minutes, or until puff pastry is golden and filling is bubbly. Let rest 5 to 10 minutes before serving.