Allow the puff pastry to thaw overnight in the fridge or set out on the countertop away from the heat to thaw. See my notes for tips on thawing puff pastry on the countertop.
Preheat oven to 400 degrees F.
Heat a large pot over medium high heat. Add the flour and butter and stir for 3 - 4 minutes until it becomes a light tan color. Next add the chicken stock, potatoes, celery, onions, cayenne pepper, and dried herbs. If using uncooked chicken, add it at this time. Bring to a boil and cook until the chicken and potatoes are cooked through. If using pre-cooked chicken, add it at this time. Taste for salt & pepper. Remove from the heat.
Stir in the frozen peas and carrots. Fill your baking dish or ramekins no more than 3/4 full and allow it to cool. The frozen peas and carrots should help speed this up.
Meanwhile, unroll your frozen puff pastry and dust it with flour. Roll it out with a rolling pin until smooth and cut out pieces large enough to drape the baking dishes leaving 1 inch extra on the sides.
Crack the egg into a bowl and mix in 1 Tablespoon of water to create an egg wash. Brush the edges and sides of the baking dish with the egg wash and drape the puff pastry on top while pressing on the sides to create a seal. Next brush the tops of the puff pastry with the egg wash to create a shiny, golden brown crust.
Bake it in the oven for 30 - 40 minutes until the puff pastry is golden brown and becomes flakey and crispy.