Melt butter in a large skillet over medium heat. Cook and stir onion, flour, salt, and pepper in melted butter until the onion is translucent, about 5 minutes.
Remove skillet from heat and pour chicken broth and milk into the skillet; bring the mixture to a boil and cook to thicken slightly, about 1 minute.
Remove skillet from heat and stir chicken, peas and carrots, and potatoes into the broth mixture.
Press one pie pastry into the bottom of a deep-dish pie pan. Pour the broth mixture into the pie pastry. Top with remaining pastry and press edges together to form a seal. Cut several slits into the top pastry. Place pie plate on a baking sheet.
Bake in preheated oven until the crust is golden brown, about 30 minutes. Let pie cool and filling thicken at room temperature for 15 to 20 minutes before cutting.