Combine flour, sugar, and salt in a large bowl. Add cubed butter and toss to coat. Roll butter into thin sheets with a rolling pin, combining with flour. Chill mixture for 15 minutes, then add buttermilk and cold water. Mix until dough just comes together.

Divide dough into two portions (one slightly larger) and shape into disks. Wrap in plastic and refrigerate for at least 30 minutes.

In a large skillet, melt butter over medium heat. Cook onions, carrots, celery, and garlic until softened, about 8 minutes.

Whisk in flour, thyme, parsley, salt, and pepper. Cook for 2 minutes, stirring constantly.

Gradually add chicken broth and heavy cream, whisking constantly. Simmer until thickened to coat the back of a spoon, about 5-7 minutes.

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

Preheat oven to 400°F. Roll out larger dough portion on a floured surface and line a 9-inch pie pan.

Pour cooled filling into the prepared crust. Roll out second dough portion and place over filling.

Trim excess dough leaving 1-inch overhang. Fold and crimp edges to seal. Cut 4-5 slits in top crust.

Brush entire top surface with beaten egg wash.

Bake for 45 minutes until crust is golden brown and filling bubbles through vents. Let cool for 10-15 minutes before serving.