Make Pie Dough. Combine White Lily flour and salt in a large bowl. Use a pastry blender to cut shortening in until mixture is crumbly. Starting at one side of the bowl and working your way across the bowl, sprinkle 1 tablespoon water over flour mixture and work it in with a fork. Sprinkle the next tablespoon beside the part you just worked in, and continue across the bowl until there is no dry flour left.
Shape dough into a flat rectangle, wrap in plastic wrap and refrigerate for at least 1 hour.
Preheat oven to 400 degrees and grease a 7x11-inch baking dish.
In a large pan, melt butter over medium heat. Add celery, carrots, and onion and cook until soft, about 5 minutes.
Sprinkle flour over vegetables and cook and stir for 1 minute.
Gradually stir in chicken broth and milk.Cook and stir until mixture is thickened and bubbly.
Add salt, pepper, thyme, and poultry seasoning.
Stir in chicken and peas and pour mixture into prepared casserole dish.
Roll dough out into a large rectangle on a lightly floured surface. Cut into 3/4-inch wide strips to fit both across the long way and the short way.
Arrange strips in lattice pattern and brush with egg.
Bake for 35 to 40 minutes or until pastry is golden.