Preheat oven to 375 degrees F. If using a bottom crust, roll out crust larger than baking dish or pie plate. Lightly grease bottom and sides of pie dish and lay bottom crust.
In a separate pan, heat 2 Tablespoons butter over a medium heat. Add onions, celery and carrots (if using raw carrots). Saute about 5 minutes.
Meanwhile, make the veloute. Melt remaining 1/4 cup butter in a large sauce pan, add flour and cook for a couple of minutes on a medium low heat, but do not allow to brown.
Slowly whisk in chicken broth, whisking constantly. Cook for 3 or 4 minutes until thickened. Add salt, pepper, poultry seasoning and fresh thyme. Add sautéed vegetables and cook until tender. About 10 minutes. Add chicken, frozen peas and cook another 3-5 minutes until all the veggies are tender and filling is hot. Add fresh chopped parsley. Check seasoning, adding more salt or pepper to taste. Pour filling in a deep dish pie plate and top with crust.
Poke holes in the top to let steam escape. Place pie on a baking sheet to catch drips.
Brush with egg wash and bake for 45 minutes to one hour at 375 degrees F., or until deep golden brown.