Preheat your oven to 400°F (200°C).
Melt butter in a large deep skillet over medium heat until it foams.
Sauté the carrots and celery in the butter until they soften significantly.
Whisk the flour into the butter and vegetables to create a thick paste.
Slowly pour in the chicken broth and heavy cream while whisking constantly to prevent lumps.
Simmer the sauce until it thickens enough to coat the back of a spoon.
Stir in the cooked chicken and frozen peas until well combined, then transfer the mixture to a pie dish or baking pan.
Roll out the puff pastry to fit your dish and drape it over the filling, crimping the edges to seal.
Brush the top of the pastry with the beaten egg.
Bake for 35 minutes or until the pastry is puffed and golden brown.