Heat oven to 400°F. Cut eight 2 ½-inch circles from puff pastry and place on parchment-lined baking sheet. Brush with egg and top each with 1 sprig dill, pressing to adhere. Bake until puffed and golden brown, 10 to 13 minutes.
Meanwhile, heat butter in large saucepan on medium. Add leeks, carrots, celery, and ¼ teaspoon salt and sauté until beginning to soften, 8 to 10 minutes.
Sprinkle in flour and cook, stirring, 1 minute. Slowly stir in broth and bring to a simmer. Simmer until vegetables are tender and soup has thickened, 10 to 12 minutes.
Remove from heat and stir in peas and chicken and its juices and heat through. Season with ¼ teaspoon pepper and stir in parsley and dill. Serve with puff pastry rounds.
