Preheat your oven to 425°F (220°C).
In a large skillet, melt the butter over medium heat.
Add the onion, flour, salt, and pepper to the melted butter. Stir and cook until the onion is translucent, about 5 minutes.
Gradually pour in the chicken broth and milk, whisking constantly to avoid lumps. Bring the mixture to a gentle boil and cook until slightly thickened, about 1 minute.
Remove the skillet from heat and stir in the cooked chicken, peas, carrots, and drained potatoes. Set aside.
Press one pie crust into the bottom of a 9-inch deep-dish pie pan.
Pour the chicken mixture into the pie crust, spreading it evenly.
Top with the remaining pastry, pressing the edges together to seal. Use a sharp knife to cut several slits in the top crust to vent steam.
Place the pie on a baking sheet to catch any drips and bake for 30 minutes, or until the crust is golden brown.
Let the pie cool for 15-20 minutes at room temperature. This allows the filling to set, making it easier to slice.