Step 1Bring the chicken stock to a boil in a deep skillet. Add the chicken breasts, reduce the heat to a simmer and poach the chicken until cooked through, about 15 minutes. Remove; when cool enough to handle, chop and set aside. Reserve the stock.
Step 2Meanwhile, melt the butter in a large saucepan over medium heat and add the carrot, onion and celery. Cook until soft, about 10 minutes. Add the flour and stir steadily for 3 minutes. Then add the stock slowly, stirring, to avoid lumps. Bring to a boil, then reduce the heat to medium-low and simmer for 15 minutes.
Step 3Raise the heat to medium and add the cream; cook for 10 minutes. Add the peas and chicken, salt and pepper to taste, and the Old Bay seasoning. Cook until hot throughout, about 5 minutes.
Step 4Heat the oven to 425 degrees.
Step 5Place the mixture in a shallow 2-quart baking dish. Lay the puff pastry sheet on top, trimming it to fit the dish. You might want to score the top for a prettier presentation. Bake until the pastry is puffed and lightly golden, 15 minutes. Combine the egg and 1 tablespoon of water and brush over the pastry. Return the pie to the oven and continue baking until puffed up and a little more golden, 2 to 3 minutes.