In a large bowl, mix 2 and 1/2 cups all-purpose flour with 1 tsp salt. Cut in 1 cup cold butter using a pastry cutter or two forks until crumbly. Add 6 to 8 tbsp cold water one spoonful at a time, mixing until the dough holds together. Divide the dough in half, wrap each piece, and chill for at least 30 minutes.

In a large skillet over medium heat, melt 3 tbsp butter. Add 1 small chopped onion, 1 cup diced carrots, and 1 cup sliced celery. Cook for about 6 minutes until softened. Stir in 1 cup frozen peas and heat through.

Sprinkle 1/3 cup all-purpose flour over the vegetables and stir to coat. Slowly pour in 2 cups chicken broth while stirring, then add 1 cup milk or half and half. Stir continuously over medium heat until the mixture thickens into a creamy sauce. Season with salt and pepper to taste.

Add 2 cups shredded or chopped cooked chicken to the sauce and simmer for a few minutes. Remove from heat and let cool slightly.

Roll out one half of the chilled dough to fit the bottom of a pie dish. Add the chicken and vegetable filling. Roll out the second half of the dough and place on top. Trim the edges and pinch to seal. Cut small slits in the top to allow steam to escape. Optionally, brush the crust with a beaten egg.

Preheat the oven to 400°F (200°C). Bake the pie on the lowest oven rack for about 35 minutes, or until the crust is golden brown. If the pie browns too quickly, cover loosely with foil. Let rest for 10 minutes before serving.