Thaw the puff pastry overnight in the fridge, or set it out on the counter away from heat until pliable. See the notes for countertop thawing tips.
Preheat oven to 400 degrees F.
Heat a large pot over medium-high heat. Add the butter and flour and stir for 3 to 4 minutes, until it turns a light tan color. Add the chicken stock, potatoes, celery, onions, cayenne pepper, and dried herbs. If using raw chicken, add it now and bring to a boil, cooking until the chicken and potatoes are cooked through. If using cooked chicken, stir it in after the potatoes are tender. Taste and season with salt and pepper. Remove from the heat.
Stir in the frozen peas and carrots. Fill each ramekin or baking dish no more than 3/4 full and let the filling cool. The frozen vegetables help speed this up.
On a lightly floured surface, roll the thawed puff pastry smooth with a rolling pin and cut pieces large enough to drape each dish with about an inch of overhang on all sides.
Beat the egg with 1 tablespoon of water to make an egg wash. Brush the edges and sides of each dish with egg wash, drape the pastry on top, and press the sides to seal. Cut a small slit in the center of each to vent steam, then brush the tops with egg wash.
Bake for 30 to 40 minutes, until the puff pastry is golden brown, puffed, and crisp. Place the dishes on a baking sheet to catch any spillover. Let stand a few minutes before serving.