Preheat your oven to 400°F (200°C) and prepare your pie crusts by unrolling them. Then, in a large saucepan, melt the butter over medium heat and whisk in the flour until a smooth paste forms, cooking for one minute.

Gradually whisk in the chicken broth and milk into the butter-flour mixture, stirring constantly until the mixture thickens. Bring the mixture to a simmer, then reduce the heat.

Stir in the shredded cooked chicken, frozen mixed vegetables, salt, and pepper into the thickened sauce, ensuring all ingredients are well combined.

Pour the chicken and vegetable mixture into one of the pie crusts, which should be placed in a 9-inch pie plate. Top with the second pie crust, crimping the edges to seal and cutting a few slits in the top for venting.

Bake the pot pie for 30-35 minutes, or until the crust is golden brown and the filling is bubbly. Allow it to cool for a few minutes before serving.