Preheat your oven to 425°F (220°C) and position a rack in the lower third. Lightly grease a 9-inch pie dish.
Dice the onion, carrots, and celery into small uniform pieces. Measure out the frozen peas and corn.
In a large skillet, melt the butter over medium heat. Add the onion, carrots, and celery and cook for 5-7 minutes until softened but still with a slight bite.
Sprinkle the flour over the vegetables and stir constantly for 1-2 minutes until absorbed and nutty smelling to cook the roux.
Slowly stir in the chicken broth and milk. Bring to a gentle simmer and cook 3-4 minutes until the sauce thickens to a gravy consistency.
Stir in the salt, pepper, garlic powder, thyme, celery seed (if using), and parsley. Taste and adjust seasoning.
Add the diced chicken, frozen peas, and corn. Mix well and remove from heat. Cool the filling for 10-15 minutes.
Unroll one pie crust and press into the prepared pie dish. Trim excess, leaving 1/2 inch overhang.
Pour the cooled filling into the crust and spread evenly.
Place the second crust on top. Trim, fold edges under, and crimp to seal tightly.
Whisk egg with water and brush over the top crust. Cut 4-5 slits for vents.
Place on a baking sheet and bake for 30-35 minutes until the crust is deep golden brown and filling is bubbling. Cover edges with foil if browning too quickly.
Remove from oven and cool on a wire rack for at least 15-20 minutes before slicing to allow filling to set.