Combine flour, sugar, and salt in a large bowl. Add the cold butter cubes and toss them in the flour mixture. Using a rolling pin, flatten each butter cube directly in the flour mixture into thin sheets about 1/8-inch thick. Place the bowl in the refrigerator for 15 minutes. Remove and add buttermilk, mixing with a fork until dough just comes together. If the mixture seems too dry and crumbly, add cold water one tablespoon at a time until it holds together. Divide the dough in half, shape into flat disks, wrap in plastic, and refrigerate for at least 30 minutes.

Melt butter in a large skillet over medium heat. Add onion, carrots, and celery, cooking for 5-6 minutes until vegetables soften. Add garlic and cook for 1 minute until fragrant.

Sprinkle flour over the vegetables and stir constantly for 1-2 minutes. Gradually whisk in chicken broth and heavy cream, stirring continuously to prevent lumps. Add thyme, parsley, salt, and pepper. Simmer for 3-4 minutes until the mixture thickens to a gravy-like consistency.

Stir in shredded chicken and frozen peas. Remove from heat and let the filling cool for 10-15 minutes.

Preheat oven to 400°F. Roll out one dough disk on a floured surface to about 12 inches in diameter. Transfer to a 9-inch pie pan, letting excess hang over the edges. Pour the cooled filling into the crust.

Roll out the second dough disk and place it over the filling. Trim excess dough, leaving about 1 inch of overhang. Fold the edges under and crimp to seal. Cut 4-5 small slits in the top crust. Brush the entire top with beaten egg.

Bake for 45-50 minutes until the crust is deep golden brown and the filling bubbles vigorously through the vents. If edges brown too quickly, cover with aluminum foil. Let rest for 10 minutes before slicing.