Start by melting butter in a large skillet or saucepan over medium heat. Once melted, stir in the chopped onion and cook for about 3–4 minutes until the onions are soft and translucent. This is where your filling starts building flavor.
Next, sprinkle the flour over the onions and butter, whisking continuously to make a smooth roux. Cook this mixture for 1–2 minutes to remove the raw flour taste. Then slowly pour in the chicken broth while whisking, making sure there are no lumps.
Add in the milk and continue to stir the mixture as it simmers. After about 5 minutes, the sauce should begin to thicken. At this point, fold in your cooked chicken and the thawed peas and carrots. Stir everything together until it’s evenly mixed. Taste and season with salt and pepper as needed.
Place one of the unbaked pie crusts into a 9-inch pie dish, gently pressing it into the bottom and sides. Make sure there are no air pockets. Spoon the creamy filling into the crust, spreading it out evenly. Try not to overfill, as it can cause the filling to spill out during baking.
Now, gently lay the second pie crust over the top. Trim any overhanging edges and crimp the sides to seal the pie. Use a sharp knife to cut 3–4 small slits in the center of the top crust. This allows steam to escape while baking and prevents the crust from getting soggy.
Finally, use a pastry brush to apply the beaten egg all over the top crust. This step gives the pie that golden, glossy finish that makes it look bakery-fresh when it comes out of the oven.
Preheat your oven to 425°F (220°C). Place the assembled pie on the middle rack and bake for about 30–35 minutes. You’ll know it’s done when the crust is a deep golden brown and you can see the filling bubbling up through the vent slits.
If the edges of your crust start to brown too quickly, you can cover them with foil or a pie shield halfway through the baking time. This helps ensure even browning without burning the outer crust.
Once baked, remove the pie from the oven and let it rest for at least 10 minutes before slicing. This allows the filling to set slightly, making it easier to serve.