Whisk together flour, sugar, and salt. Add cold butter and coat in flour. Flatten into flakes, chill 15 minutes, then mix in cold buttermilk. Add water as needed and form two disks. Chill.
Melt butter in a skillet, add onion, carrots, and celery. Cook 5–7 minutes until soft. Add garlic and cook 1 minute more.
Sprinkle flour over vegetables and stir for 2 minutes. Gradually add broth, then heavy cream. Stir in herbs, salt, and pepper. Simmer until thickened.
Add chicken and peas to sauce and stir. Cool slightly before filling crust.
Preheat oven to 400°F (200°C). Roll out crust, line pie pan, and add filling. Top with second crust. Trim, seal, and crimp edges. Brush with egg wash.
Bake 45 minutes or until golden. Cool 10–15 minutes before slicing.
