Lay out the box of refrigerated dough and allow to come to room temperature.
Melt butter over low heat.
Stir in flour to make a smooth paste. This is called a roux.
In separate pan, bring chicken broth to a boil. Lower heat to medium high.
Add roux, a tablespoon at a time to the chicken broth, continually whisking
Keep adding roux and whisking until the mixture is thickened.
Add milk, a little at a time and continue stirring over heat until mixture is smooth and a pudding like consistency. Do not allow to get too thick as it will continue to thicken as it bakes in crust. You can add more milk if it becomes too thick.
Add chicken, mixed vegetables, thyme, salt and poultry seasoning. Stir and set aside.
Unroll pie dough onto floured surface.
Roll dough out lightly.
Using one of the dishes you are baking in as a template, cut the shape of the bottom of the dish out of the pie dough. You may need to gather pieces and reroll dough as you cut.
Place a piece in the bottom of each dish.
Cut pieces out of dough to fit the top of each dish plus one extra inch all the way around.
Fill each dish about ¾ full with the chicken filling.
Mix egg and 1 T. milk with a fork, whipping lightly.
Brush around edges of each top piece.
Flip pieces over onto the tops of the filled dishes
Fold top pieces under about 3/4 inch onto itself and crimp the edges. Brush the surface of the tops with egg wash. Cut several slits in tops to allow steam to escape.
Bake on center rack of oven for approx. 30-35 min. or until golden brown.
I made this recipe into three mini pies. However, you can make them into one large 9 inch pie if you would like.