Cut the puff pastry into eight squares, brush with beaten egg and bake at 400°F for 15 minutes until deeply puffed and 160°F inside.
Melt the butter in a Dutch oven and cook the onion, carrots and celery for 7 minutes.
Stir in the flour for 2 minutes, then whisk in the broth and milk.
Add the salt and black pepper and simmer for 12 minutes until the broth coats a spoon.
Add the chicken and peas and cook for 6 minutes until the chicken reaches 165°F.
Divide the soup among bowls and set one pastry square on each just before serving.