Melt the butter in a small saucepan over medium heat. Once butter is melted, whisk in flour until smooth. Slowly pour in the chicken broth and whisk until the sauce has thickened. Add the milk, salt, and pepper and whisk until the sauce is bubbling and thick enough to coat the back of the spoon.
To assemble the pie: Press one of the pie crusts into the bottom and sides of a 9 inch pie plate. Spread the cooked turkey or chicken on the base and top with the mixed vegetables. Pour the gravy over the top of the chicken and vegetables. Place the top crust over the pie. Trim and crimp the edges of the crust together. Using a paring knife cut slits across the top of the pie.
Bake the pot pie at 375 degrees F for 40 minutes or until the crust is golden brown and the filling is bubbling. Let pie cool for 10 minutes before serving.