Preheat oven to 350 degrees.
In a medium sauce pan, melt 3 tablespoons of butter. Stir the 3 tablespoons of flour into the melted butter until a paste forms (a roux). Cook the paste, stirring for 1 minute (this will take the raw flour taste out of the gravy).
Slowly add the chicken broth into the roux. Over medium heat, whisk the mixture 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 bite sized chicken pieces. Cool the filling while preparing the crust.
Roll out the thawed pie crusts out on a lightly floured surface to enlarge them just slightly.
Using an inverted bowl and a paring knife cut 3 - 6" circles from each piece of pie dough.
Place approximately 1½ tablespoons of 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, as this will 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.