Preheat oven to 425 degrees. Place a 1 ½-cup ovenproof dish on pastry on a floured surface. Cut circle from pastry, ½ inch larger than dish; cut vents. Repeat. Refrigerate on a parchment-lined baking sheet.
Melt butter in a large skillet over medium heat. Add onion. Cook, stirring, until soft, 4 minutes. Add carrot, celery, and potato. Cook, stirring, until soft, 6 minutes. Stir in flour, then broth; bring to a boil. Add greens and chicken. Simmer until thickened, 2 minutes; season.
Divide between dishes. Top with pastry, press edges to seal, and brush with egg. Bake on a baking sheet until golden, 25 minutes.
