Preheat oven to 400ºF (200ºC).
Melt butter on medium to low heat in a 10-inch cast iron skillet. Add onions, carrots, celery and potatoes to pan and cook for 4-5 minutes until they begin to soften. Add flour and continue to stir until well blended and the vegetables are coated. Add broth and continue stirring until the flour is fully integrated and a sauce has begun to form.
Stir in chicken, herbs and seasonings and continue cooking until it starts to bubble and the potatoes are pierced easily with a fork, about 7-8 minutes. If you find the pan a little dry, stir in ¼ cup filtered water. Fold in the peas and remove pan from stove.
Top with puff pastry, folding any excess pastry under the edges so the pan rim is exposed. Cut 2 or 3 vents into the crust to allow steam to escape and brush with egg wash.
Bake for 35-40 minutes until the crust is nicely browned. Let cool for 10 minutes and serve.