Mix the 2 1/2 cups flour with 1 teaspoon salt, cut in the shortening until pea-size pieces remain, and stir in the ice water 1 tablespoon at a time until the dough holds together.
Divide the dough into 2 disks, wrap, and refrigerate for 30 minutes.
Melt the butter in a deep skillet, cook the onion, carrots and celery for 8 minutes, then stir in the 1/2 cup flour for 1 minute.
Whisk in the broth and milk, simmer until thick, fold in the chicken, peas, thyme, pepper and remaining salt, and cool the filling for 20 minutes.
Heat the oven to 400°F, roll one disk into a 12-inch circle, and fit it into a 9-inch deep-dish pie plate.
Add the filling, cover with the second rolled crust, crimp the edges, cut 4 steam vents, and brush with the egg wash.
Bake for 45 to 50 minutes until the crust is deep golden and the filling bubbles through the vents, then rest for 20 minutes before cutting.