Preheat your oven to 350°F.
In a 10-inch cast iron skillet, heat the butter and olive oil over medium heat until melted. Add the onion, celery, and carrots and cook for about 3 minutes until the onion is soft and translucent.
Sprinkle the flour over the vegetables and stir constantly for 1 minute. Slowly stir in the chicken broth until combined, then add the cream. Stir in the thyme, poultry seasoning, salt, and peppers.
Add the chicken, potato, peas, and parsley, carefully stirring to combine. Remove the skillet from the heat.
Carefully cover the skillet with the puff pastry (or pie dough), then tuck in the edges. Brush the top of the dough with the beaten egg, then cut a few slits in the center of the pie.
Bake for 35-45 minutes, until the crust is golden brown and the filling is bubbling around the edges. Allow the pie to cool for at least 10 minutes before slicing and serving warm.