Melt the butter in a saucepan over medium-low heat, and cook and stir the onion with butter until translucent, about 5 minutes. Stir in the flour, salt, and pepper to make a paste. Whisk in the chicken soup and milk, and cook and stir the sauce until smooth and thickened, about 5 minutes. Lightly mix in the cooked chicken and mixed vegetables, and pour the chicken and sauce into a 2-quart deep casserole dish.
Unroll the pie crust, lay it on the casserole dish, and fold and press the edges of the crust down to seal the crust to the dish. Cut several small slits in the crust.
Bake in the preheated oven until the crust is golden brown and the filling is bubbling, 35 to 40 minutes. Cool 10 minutes before serving.