Prepare the chicken by shredding or dicing it into bite-sized pieces.
In a large skillet, heat olive oil over medium heat. Add diced onion and celery, sauté for 5 minutes until softened. Stir in frozen mixed vegetables and cook for another 2-3 minutes.
Sprinkle gluten-free flour over the vegetable mixture, stir well, and cook for about 1 minute. Gradually pour in chicken broth and milk, stirring constantly until thickened (3-5 minutes). Add shredded chicken, thyme, garlic powder, salt, and pepper. Mix until well combined and heated through.
Prepare the pie crust by rolling it out and fitting it into a 9-inch pie dish.
Pour the chicken filling into the prepared pie crust, spreading it evenly. If using a second pie crust, roll it out and place it over the filling. Seal the edges and cut slits in the top crust.
Brush the top crust with the beaten egg for a golden finish.
Preheat oven to 425°F. Bake the assembled pie for 30-35 minutes until the crust is golden brown.
Let the pie cool for 10 minutes before serving.