In a large saute pan or deep pot warm some oil over medium heat. Then add the diced onion, celery, and minced garlic. Saute for a few minutes.
Add the frozen vegetable mix along with potatoes if using. Stir and continue to cook until the veggies start to soften.
Sprinkle with the arrowroot or cornstarch and mix well.
Slowly stir in the chicken stock and thyme and bring to a simmer. You will notice it start to thicken up at this point.
Once the veggies are tender add the diced chicken or turkey.
Stir in the milk, salt and pepper. Let it simmer for about 5 minutes for the flavors to blend and the sauce to thicken.
Taste and add salt and pepper to your preferance. Remove from heat.
Grease or butter a 10-inch deep-dish pie plate or 11x 7 baking pan and place on top of a baking sheet to catch spills. Set aside.
Pour the filling into the pie plate.
Roll out the pie crust and put it over the filling. Make a few slits in the crust to allow steam to escape. Brush the crust with a lightly beaten egg.
Bake for 45 -50 minutes or until the crust is brown and the filling is bubbly.