Preheat oven to 425°F (220°C).
In a large skillet or saucepan, melt butter over medium heat. Add carrots, celery, and peas. Cook for about 5 minutes, stirring occasionally.
Stir in flour, salt, pepper, garlic powder, onion powder, and thyme until vegetables are well coated.
Gradually whisk in chicken broth and milk. Simmer and stir until the mixture thickens, about 5–7 minutes.
Add cooked chicken and stir to combine. Remove from heat.
Roll out one pie crust into a 9-inch pie dish. Pour the hot chicken filling into the crust.
Cover with the second pie crust. Seal and flute the edges. Cut slits in the top to allow steam to escape.
Brush the top crust with a beaten egg for a golden finish.
Bake for 30–35 minutes, or until the crust is golden brown and filling is bubbling.
Let rest for 10 minutes before serving.