Preheat oven to 400°F (200°C).
In a skillet, melt butter over medium heat. Add flour and cook, stirring constantly, for about 1 minute.
Gradually whisk in chicken broth and milk. Cook until the mixture thickens, about 5 minutes.
Add shredded chicken and frozen peas and carrots. Season with salt and pepper. Stir to combine.
Pour the mixture into a baking dish.
Roll out the puff pastry and place it over the filling. Trim any excess and crimp the edges to seal.
Brush the puff pastry with the beaten egg.
Cut a few slits in the top to allow steam to escape.
Bake for 30-35 minutes, or until the pastry is golden brown.
Let cool for a few minutes before serving.