When the dough has chilled, preheat the oven to 425° F. Steam the carrots and celery for 5 minutes in a steamer. Add the onions and continue to steam for an additional 10-12 minutes or until the carrots are tender.
In a separate large saucepan, melt the butter over medium heat, add the flour and whisk together until smooth. Add the chicken broth and milk and continue stirring over high heat until the mixture comes to a boil. Cook for an additional minute or so until thick, then reduce heat to low.
Add the steamed vegetables and peas to the sauce and simmer the mixture over medium/low heat for 4 to 5 minutes.
As the filling simmers, roll out the dough on a floured surface. Use the dish/es you plan to bake the pie/s in as a guide for cutting dough. The filling will fit four 16 oz casserole dishes. Cut the dough about a half-inch larger then the dish (Note: I use a 9x13 sheet pan, or if I'm feeling fancy, I'll use ramekins for individual servings--the choice is up to you).
Spoon the chicken and vegetable filling into each dish and cover with the dough, sealing the edges. Brush some beaten egg or butter on the dough of each pie (optional).