Prepare the Dough – Whisk flour, sugar, and salt. Add butter cubes, flatten with rolling pin, then chill 15 minutes. Add buttermilk and form a shaggy dough. Divide and chill.
Cook the Vegetables – Melt butter, sauté onion, carrots, and celery for 7 minutes. Add garlic and cook 1 minute.
Make the Sauce – Add flour, stir 2 minutes, whisk in chicken broth, then add heavy cream, thyme, parsley, salt, and pepper. Simmer until thick.
Combine Everything – Stir in shredded chicken and peas. Remove from heat.
Assemble the Pie – Roll dough, line pie dish, fill with mixture, top with second crust, crimp edges, and brush with egg wash.
Bake to Perfection – Bake at 400°F (200°C) for 45 minutes until golden brown.
Rest and Serve – Cool for 10 minutes before slicing.
