Preheat the oven to 425 degrees F (220 degrees C).
Melt butter in a 2-quart saucepan over medium heat. Stir in flour, onion, salt, and pepper. Cook, stirring constantly, until mixture is bubbly, 2 to 3 minutes. Stir in chicken broth and milk; bring to a boil, stirring constantly. Boil and stir for 1 minute. Stir in chicken, peas, and carrots and remove from the heat.
Roll both sheets of puff pastry into 12-inch squares. Ease 1 sheet into the bottom of an ungreased 9-inch square pan, then pour chicken mixture on top. Place remaining pastry over the chicken mixture. Turn pastry edges under and flute.
Bake in the preheated oven until golden brown, about 35 minutes.