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In a large bowl, whisk together flour, sugar, and salt. Add cold butter cubes and toss to coat. Using a rolling pin, press the butter into thin flakes and mix gently. Chill for 15 minutes, then add cold buttermilk. Add cold water as needed to form a shaggy dough. Divide into two disks and refrigerate.
Melt butter in a skillet over medium heat. Add onion, carrots, and celery. Cook 5–7 minutes until softened. Add minced garlic and cook 1 minute more.
Sprinkle flour over vegetables and stir continuously for 1–2 minutes. Gradually pour in chicken broth, whisking until smooth. Add heavy cream, thyme, parsley, salt, and pepper. Simmer 8–10 minutes until thickened.
Add shredded chicken and frozen peas to the sauce. Stir until evenly coated, then cool slightly.
Preheat oven to 400°F (200°C). Roll one dough disk into a 12-inch circle and fit it into a 9-inch pie dish. Add filling. Roll second disk, cover, trim, and crimp edges. Cut small slits for steam vents and brush with beaten egg.
Bake for 45 minutes until golden brown and crisp. Cover edges with foil if browning too quickly. Let cool 10–15 minutes before serving.