Preheat the oven to 400°F (200°C).
In a saucepan, melt butter and whisk in flour until smooth.
Gradually add chicken broth and milk, stirring until thickened.
Mix in chicken, peas, carrots, salt, and black pepper.
Place one pie crust in a pie dish, add filling, and cover with the top crust.
Seal edges, cut slits for steam, and bake for 30–40 minutes until golden brown.
Let cool slightly before serving.