Start by dicing the chicken breasts into bite-sized pieces. Chop the carrots, potatoes, and onion to a similar size to ensure even cooking. If using frozen peas, rinse and drain them ahead of time.
In a large skillet over medium heat, melt 1 tablespoon of butter. Add the diced chicken and season lightly with salt and pepper. Cook until the chicken is just cooked through, about 5-6 minutes. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining butter and sauté the onions, carrots, and potatoes until the onions turn translucent and the vegetables start to soften, about 5-7 minutes.
Sprinkle the flour over the vegetables and stir constantly for 2 minutes to cook the raw flour taste out. This step is essential to thicken the sauce later.
Slowly whisk in the chicken broth and milk, ensuring no lumps form. Add the dried thyme, garlic powder, salt, and pepper. Bring the mixture to a gentle simmer; it will thicken as it heats.
Return the cooked chicken to the skillet and stir in the peas. Cook for another 2-3 minutes until the peas are warmed through and the filling is thick and creamy.
Preheat your oven to 425°F (220°C). Roll out one pie crust and fit it into your pie dish. Pour the chicken and vegetable filling evenly into the crust. Cover with the second crust, trimming any excess dough. Crimp the edges to seal.
Brush the top crust with the beaten egg for a beautiful golden finish. Cut a few small slits in the top crust to allow steam to escape during baking.
Place the pie on a baking sheet and bake for 30-35 minutes or until the crust is golden brown and bubbling. Let it rest for 10 minutes before serving.
