Preheat the oven to 425 degrees F (220 degrees C).

Melt butter in a large skillet over medium heat. Cook and stir onion, flour, salt, and pepper in hot butter until onion is translucent, about 5 minutes.

Pour in broth and milk; bring to a boil and cook until slightly thickened, about 1 minute. Remove the skillet from heat and stir in chicken, peas and carrots, and potatoes.

Press 1 pie pastry into the bottom of a 9-inch, deep-dish pie pan. Pour chicken mixture into pie pastry. Top with remaining pastry and press edges together to seal. Cut several slits in top pastry. Place pie on a baking sheet.

Bake in the preheated oven until crust is golden brown, about 30 minutes. Let pie cool and filling thicken at room temperature for 15 to 20 minutes before cutting.