Preheat oven to 425°F (220°C).
In a large skillet, melt butter over medium heat. Add carrots and celery, cooking 5 minutes until softened.
Stir in flour and cook 1 minute, then gradually whisk in chicken broth and milk.
Add salt, pepper, and thyme, simmering until thickened (3–4 minutes).
Stir in chicken and peas; remove from heat.
Line a 9-inch pie dish with one crust, fill with mixture, and top with second crust. Seal and crimp edges; cut small slits for steam.
Brush with egg wash and bake for 30–35 minutes, until golden brown.
Let rest 10 minutes before slicing.
