Mix flour and salt together and then cut in the shortening with a pastry blender or 2 forks.
Add in one tablespoon water at a time, gently mixing the dough between each addition. Add water until dough can be pressed together into a ball.
Divide dough in two. Form into discs and wrap in plastic and place in the fridge. Meanwhile, make the filling.
CHICKEN FILLING
Preheat oven to 425 degrees F.
Place potato and carrots in a pot, cover with water and bring to a boil. Cook for 7 to 8 minutes or until the vegetables are crisp tender. Drain and set aside.
In a large skillet, heat butter over medium heat. Add the onion, celery, and garlic and cook until translucent and tender, about 5 to 8 minutes.
Stir in the seasonings and flour and cook for 2 minutes while stirring. Whisk in the broth a little at a time, followed by the milk. Cook for 2 to 3 minutes or until thickened. Stir in the chicken, potato, and carrots. Set aside. (Mix in the frozen peas just before adding to the pie plate.)
ASSEMBLY AND BAKING
Pull one disc out of the fridge and roll out, large enough to place in the pie plate. Gently lift and press into a 9 or 10 inch pie plate. Roll out the other disc for the top.
Spoon the filling into the pie plate on top of the crust. Gently place the remaining rolled out pie dough on top. Fold the edges of the top crust over the bottom and then pinch to seal. Finish the edges by fluting or smashing with a fork. Make 6 to 8 slits in the top so the steam can vent.
Whisk the egg and water together and brush over the top and edges of crust.
Bake for 35 to 38 minutes or until golden brown. Let cool 5 to 10 minutes before slicing.