In a large mixing bowl, combine all-purpose flour, sugar, and salt. Cut in cold unsalted butter until mixture resembles coarse crumbs. Stir in cold buttermilk; form into a ball, divide into two disks, wrap, and refrigerate for at least 30 minutes.

In a large skillet, melt unsalted butter over medium heat. Add diced onion, sliced carrots, sliced celery, and minced garlic; sauté for 5-7 minutes until tender. Stir in all-purpose flour, thyme, parsley, salt, and pepper; cook for 1-2 minutes until thickened. Gradually add chicken broth and heavy cream, stirring continuously for about 5 minutes until thickened.

Remove the skillet from heat and stir in shredded chicken/turkey and frozen peas. Adjust seasoning with salt and pepper if desired. Set aside to cool while preparing the crust.

Preheat your oven to 400°F. Roll out one disk of dough into a 12-inch circle; place into a 9-inch pie pan. Pour chicken filling into the crust. Roll out second disk, drape over filling, seal edges, and cut slits for steam release.

Brush the top crust with egg wash and place assembled pie on a baking sheet. Bake for 45 minutes or until crust is golden brown and filling bubbles up through slits.

Once baked, let Chicken Pot Pie cool for about 10 minutes before serving.