In a large bowl, combine flour, sugar, and salt. Add cubed butter and toss to coat. Transfer to a clean surface and roll butter into thin sheets using a rolling pin, gathering the mixture with a bench scraper as needed until flaky.

Return mixture to the bowl and freeze for 15 minutes to chill the butter.

Remove from freezer and add buttermilk. Stir with a spoon, then gently bring together with hands. Add cold water 1 tablespoon at a time if needed. Divide dough into two disks, wrap, and refrigerate.

In a large skillet over medium-high heat, melt butter. Add onion, carrots, celery, and garlic. Cook until vegetables are tender.

Whisk in flour, thyme, parsley, salt, and black pepper. Slowly add chicken broth and heavy cream, whisking until smooth. Reduce heat and simmer for 10 minutes until thickened.

Stir in shredded chicken or turkey and frozen peas. Remove from heat.

Preheat oven to 400°F (200°C).

Roll out one dough disk into a 12-inch circle about ¼ inch thick. Transfer to a 9-inch pie pan and trim excess dough.

Fill pie crust with prepared filling.

Roll out second dough disk and place over filling. Trim edges, seal and crimp, cut slits in top, and brush with beaten egg.

Bake for 45 minutes until crust is golden brown. Let cool for 10 minutes before slicing and serving.