Preheat oven to 425 degrees F (220 degrees C). Place a baking sheet on the a rack on which you'll bake the pie.
Combine the carrots, celery, peas, green beans, corn, onion, red potatoes, and chicken broth in a large pot. Season the vegetable mixture with thyme, salt, and black pepper; bring to a boil. Reduce heat to medium-low, cover, and simmer until the vegetables are tender, about 15 minutes. Drain the vegetables and set aside.
Place dry chicken gravy mix into another saucepan and gradually whisk in the water until smooth. Bring the mixture to a boil, reduce heat to medium-low, and simmer until thickened, about 1 minute. Set gravy aside and allow to continue to thicken as it cools.
Press one of the pie crusts into the bottom of a 9-inch pie dish. Spoon a layer of gravy (about 1/3 cup) into the crust. Layer the cooked vegetables and shredded chicken into the crust until the filling is level with the top of the pie dish. Pour the rest of the gravy slowly over the filling until gravy is visible at the top. Scatter butter pieces over the filling; top with second crust. Seal the 2 crusts together and crimp with a fork.
Whisk the egg and milk together in a bowl, and brush the egg wash over the top crust. Pierce the top crust several times to vent steam.
Bake the pot pie in the preheated oven until the filling is bubbling and the crust is golden brown, about 45 minutes to 1 hour. Check periodically to see if the edge of the crust is browning too quickly; if needed, cover the edge with a strip of aluminum foil to prevent over-browning.