Heat oven to 400. Heat oil in lg oven-safe skillet over medium heat. Add onion, celery and garlic. Cook until softened. Stir in carrots, green beans and potatoes. Add chicken broth. Bring to a simmer. Cook 10 mins.
Season w/ salt and pepper. Sprinkle flour over veggies. Stir and add chicken. Bring to a simmer. Meanwhile unfold puff pastry on a floured surface. Roll out to fit skillet.
In small bowl, whisk together the egg and 1 tbsp water. Remove skillet from heat. Place pastry over skillet, tuck in edges. Brush pastry w/ egg wash. Bake in oven 15 mins or until pastry is golden.