Preheat the oven to 400°F (200°C).
Melt butter in a large skillet over medium heat, then add onion, carrots, and celery. Cook for about 5 minutes until softened.
Stir in flour and cook for 1 minute, then gradually whisk in chicken broth and milk until thickened, about 3–5 minutes. Season with salt, pepper, garlic powder, and thyme. Stir in chicken, peas, and potatoes, then remove from heat to cool slightly.
In a bowl, mix flour and salt. Cut in cold butter until crumbly, then add ice water a tablespoon at a time until the dough just comes together.
Divide into two disks and roll out on a floured surface. If too sticky, add a little flour; if dry, add a few drops of water.
Line a 9-inch pie dish with one crust and pour in the filling. Cover with the second crust, trim edges, crimp to seal, and cut small slits on top for steam.
Brush with egg wash and bake for 30–35 minutes until golden brown. Let rest 10 minutes before slicing and serving.