1. Combine the flour, sugar, and salt in a large bowl.

2. Add the cold butter cubes and work them into the flour until the mixture becomes flaky with visible pieces of butter.

3. Place the flour and butter mixture in the freezer for 15 minutes.

4. Remove the bowl from the freezer and add the cold buttermilk.

5. Mix until the dough begins to hold together. Add cold water 1 tablespoon at a time only if the mixture feels too dry.

6. Divide the dough into 2 equal portions. Shape each portion into a disk, wrap it, and refrigerate while preparing the filling.

7. Melt 1/4 cup butter in a large skillet over medium heat.

8. Add the onion, carrots, celery, and garlic. Cook until the vegetables become tender, stirring occasionally.

9. Whisk in the flour, thyme, parsley, salt, and black pepper.

10. Gradually whisk in the chicken broth and heavy cream until no flour lumps remain.

11. Reduce the heat and simmer the sauce for about 10 minutes, or until it thickens.

12. Stir in the shredded chicken or turkey and the frozen peas.

13. Remove the skillet from the heat and allow the filling to cool slightly.

14. Preheat the oven to 400 degrees Fahrenheit.

15. Roll one chilled dough disk into a 12-inch circle on a lightly floured surface.

16. Fit the dough into a 9-inch pie pan or baking dish and trim away the excess.

17. Spoon the creamy chicken filling evenly into the crust.

18. Roll the second dough disk into a circle and place it over the filling.

19. Trim and seal the edges with your fingers or a fork.

20. Cut several small vents in the top crust.

21. Brush the top and edges with the beaten egg.

22. Bake for 45 minutes, or until the crust becomes crisp and golden brown.

23. Cover the edges with foil or a pie shield if they brown too quickly.

24. Let the casserole rest for 10 minutes before slicing and serving.