In a medium sauce pan, melt 3 tablespoons of butter over medium heat. Stir the flour into the melted butter until a paste forms (a roux). Cook the roux, stirring for 1 minute (this will take the raw flour taste out of the gravy).
Slowly whisk the chicken broth into the saucepan with the roux.Continue cooking and whisking occasionally until it thickens to a chicken gravy consistency (this will take approximately 3-4 minutes).Once thickened, season the gravy with salt, pepper and poultry seasoning.
Remove the chicken gravy from the heat and stir in the thawed mixed vegetables and the prepared chicken. Set aside to cool.
Roll the thawed flat pie crusts out on a lightly floured surface to enlarge them slightly.Using an inverted bowl and a paring knife cut three 5" - 6" circles from each round of pie dough.
Place approximately 1½ tablespoons of cooled chicken filling on one half of each pie crust circle, making sure to leave about a ½" border around the circle.
Using a finger or pastry brush, run some water around the edges of each circle, to act as glue to adhere the two pie crusts edges together when pressed.Bring the top of the crust up and over the filling and press the edges firmly to seal.Use the tines of a fork to enforce the seal around the edges of each pie pocket.
Transfer the pie pockets to baking sheets lined with parchment paper (optional) and/or sprayed with cooking spray.
In a small bowl, mix the egg and water together and brush the egg wash on top of each pie pocket.
Use a paring knife to cut a few steam slits on the top of each pie pocket.
Bake the pie pockets in a preheated oven for 30-35 minutes or until the pie crust is golden brown.
Allow the pockets to cool slightly and serve.
Notes
May be frozen before baking without egg wash. Flash freeze in a single layer until firm. Wrap each partially frozen pie pocket in plastic wrap or add them in airtight freezer bags. Label and freeze. Freeze for up to 2 months.Thaw before baking. Brush with egg wash and bake (uncovered) in a moderate 350 degree oven until golden brown and hot (approximately 15 minutes, if thawed).Bake from frozen (unbaked pie pockets): Place the frozen unbaked pie pockets on a parchment lined baking sheet. Brush the tops of the pockets with egg wash. Loosely cover the pockets with foil and bake in a moderate 350 degree oven for 35-40 minutes, removing the foil for the last 5-6 minutes of baking.May be frozen after baking. Cool completely. label and date. Store in an airtight freezer bag or freezer container and freeze for up to 2 months. Thaw in the refrigerator overnight and reheat in a moderate oven or air fryer until hot.