Start by melting butter in your skillet over medium heat. Stir in onions, celery, carrots, and potato, sautéing until the onions turn translucent and veggies soften. Next, sprinkle the flour and seasonings on top, stirring well for a couple of minutes—this cooks out the raw flour taste and starts the roux.
Pour in milk, broth, and cream while stirring constantly. Add frozen peas and diced chicken. Bring the mixture to a simmer and let it cook until the sauce thickens—about 5–10 minutes. It should coat the back of a spoon; too runny and the crust will get soggy.
Cut puff pastry circles slightly larger than your ramekin tops. Spoon the thick filling into each bowl, then gently lay the pastry over, pressing edges to seal. Whisk the egg and water, brush over pastry, and place bowls on a baking sheet.
Bake at 400 °F until the pastry is deep golden, about 15 minutes. If the edges brown too fast, tent with foil and bake another 15 minutes or until the filling bubbles up around the crust edge. Let pies cool briefly before serving—they’re hot!