Make crust by putting the cubed butter, flour and salt in a food processor. Pulse until a coarse meal forms. Then add 1 tablespoon of ice water at a time until little balls start to form with the dough. It should start to all form together a little. Then form it into a ball and transfer to plastic wrap. Shape into a disk shape, and wrap in plastic wrap. Place in fridge for at least one hour, or up to a day.
Preheat oven to 375 degrees. In a large pot, melt the 6 tablespoons of butter. Add in the potatoes, carrots, onion, celery and garlic. Saute until potatoes are fork tender, about 6 minutes. Then add in the flour. Stirring to fully coat. At this point, it would be a good time to take the pie crust out of the refrigerator to soften a little.
Add in the chicken stock and bring to a boil. Let it boil until it thickens, about 2 minutes. Then add in the peas, 1/2 cup of cream, parsley, salt and pepper. Let cook for 2 more minutes before adding the shredded chicken.
Pour into an ungreased 7x11 baking dish and sprinkle with a little more salt and pepper.
Take your pie crust and place it on a floured work surface. Roll out your crust to fit a 7x11 dish, about 1/8" thick. Place on top of filling and fold the overhang under. Crimping the edges however you like.
Cut slits in the crust to allow for ventilation. Beat the egg and remaining 1 tablespoon of heavy cream together and brush over the crust. Place dish on a rimmed baking sheet to catch any overflow.
Bake for 45-50 minutes until top is golden brown. Let rest 10 minutes before serving.