Make pie crust. Place flour and salt in a food processor and pulse to combine. Scatter chilled butter pieces on top. Pulse 10 to 12 times or until mixture resembles coarse crumbs. Add cheese and pulse to combine.
Transfer mixture to a large bowl. Sprinkle 5 tablespoons of ice water on top. Stir and press the dough together using a rubber spatula. If dough is not coming together, use a little more ice water.
Turn dough out onto a piece of plastic wrap. Flatted into a disk, wrap well and refrigerate for at least 1 hour and up to 24 hours.
Preheat oven to 400 degrees.
To make filling, melt butter in a medium (10-inch) cast iron skillet over medium heat. Add onion, carrot, and celery and cook 5 minutes.
Sprinkle flour on top. Cook and stir for 1 minute.
Gradually add chicken broth whisking it into the flour well. Whisk in half and half.
Add salt, pepper, thyme, and poultry seasoning. Let mixture simmer for a few minutes.
Stir in peas and chicken. Remove from heat.
Remove dough from refrigerator and let it sit on the counter for 10 minutes.
Sprinkle some flour on a piece of parchment paper. Place dough on paper and roll it into a circle that's about 1/8-inch thick. Slide your hands under the parchment paper and transfer the pie dough to the pan, flipping it over on top of the pie filling and peeling the paper off of it. Trim excess dough or tuck it under the so that the crust fits the pan. Brush top with egg. Cut a few slits in the dough to allow steam to escape.
Place in oven and bake 30 to 35 minutes, until pie crust is lightly browned and no longer feels doughy.