Make the Pie Crust:
Combine flour, sugar, and salt in a large bowl. Add cubed butter and toss to coat. Roll butter into thin sheets using a rolling pin, scraping as needed with a bench scraper until combined into a flaky mixture. Freeze for 15 minutes.

Add Liquid:
Mix in buttermilk and stir until dough forms. If dry, add water 1 tablespoon at a time. Divide dough in two, shape into disks, wrap in plastic, and refrigerate while preparing the filling.

Prepare Filling:
In a large skillet, melt butter over medium-high heat. Add onion, carrots, celery, and garlic; cook until tender. Stir in flour, thyme, parsley, salt, and pepper. Gradually whisk in chicken broth and heavy cream until smooth. Simmer 10 minutes until thickened. Stir in shredded chicken and peas. Remove from heat.

Assemble Pie:
Preheat oven to 400°F (200°C). Roll out one dough disk to a 12-inch circle (¼-inch thick) and place in a 9-inch pie pan. Add filling. Roll out the second disk and cover pie, trimming excess dough. Seal and crimp edges, then cut slits on top for steam. Brush with beaten egg.

Bake:
Bake for 45 minutes, or until the crust is golden brown. Use a pie crust shield to prevent over-browning.

Cool and Serve:
Allow to cool for 10 minutes before slicing. Serve warm and enjoy!