Preheat oven to 400F degrees.
In a food processor, add flour, sugar, xanthan gum and salt. Pulse to combine. Add cold diced butter to the food processor and pulse until mixture resembles coarse crumbs.
In a small bowl, add egg, cold water and apple cider vinegar. Use a fork to beat the egg and mix. Pour egg mixture into food processor and pulse a few more times until dough is just moistened.
Transfer the pie dough to a large bowl and use clean hands to shape it into a ball, slightly flattened. Cover bowl with plastic wrap and place in the fridge to chill while preparing next steps.
While the dough is chilling, add oil and drained mixed vegetables to a large sauté pan on medium heat. Add chicken broth, milk, onion powder and salt and pepper. Allow to simmer for 5-7 minutes, stirring occasionally.
Mix cornstarch and warm water in a small bowl. Pour the slurry into sauté pan and stir to combine. Increase heat to high and allow liquid to come to a boil, then turn off the heat and stir in the shredded chicken. Allow the mixture to cool on the stove while following next steps.
Remove chilled pie dough from the refrigerator and use a knife to cut the dough in half. Place half back in the refrigerator and transfer the other half of pie dough to a lightly floured piece of parchment paper. Sprinkle a little flour on top of the dough and add an additional piece of parchment paper on top. Roll out the dough with a rolling pin into a 12 inch circle.
Transfer the dough to an ungreased pie pan and press into place - flattening the crust against bottom and sides as well as touching up any cracks. Pour chicken and vegetable filling over pie crust and set aside while preparing second pie crust, repeating the same process.
Cover mixture with second pie crust. Pinch and seal the edges - trimming excess crust as needed.
Optional: To give the crust a golden brown color, add an egg and a splash of water to a small bowl. Beat with a fork and brush over the top and edges of the crust.
Use a knife to cut four slits in the center of the pie. Place the pie on a baking sheet and bake in preheated oven for 30 minutes or until golden brown. Allow pie to cool for 5-10 minutes before serving.