Process the dough. Combine the flour and kosher salt in a food processor, pulse in the cold butter until pea sized, then add the ice water and vinegar and pulse until the mixture looks like damp crumbs. Rest the dough. Shape the crumbs into a ball on plastic wrap, split in two, flatten into disks, and refrigerate 1 hour.
Prepare the thickeners. Mash 4 tablespoons softened butter into 1/4 cup flour, and dissolve the cornstarch in the milk.
Cook the base. In 2 tablespoons butter, soften the onion, carrot, and celery for 3 minutes, then pour in the broth, add the potato, cover, and simmer 10 minutes.
Finish the filling. Whisk in the butter paste, then the milk mixture, boiling gently until thick, and fold in the chicken, peas, salt, and pepper. Cool 20 minutes.
Shape the pie. Heat the oven to 400 degrees F, roll the larger disk to 13 inches, fit it into a 9 inch dish, and add the cooled filling.
Bake. Cover with a 10 inch circle of dough, crimp, vent, brush with the yolk wash, and bake about 50 minutes until golden and 165 degrees F in the center. Rest 10 minutes.