Combine flour and salt in a large bowl, then cut in the cold butter until the mixture resembles coarse crumbs.

Gradually add ice water, mixing until the dough just comes together; divide in half, flatten into discs, wrap, and chill for at least 30 minutes.

Melt butter in a large pot or Dutch oven over medium heat, then add onion, carrots, and celery and cook until softened.

Stir in flour and cook for one minute, then gradually whisk in chicken broth and milk until thickened and smooth.

Remove from heat and stir in cooked chicken, peas, parsley, salt, and pepper.

Preheat oven to 400°F (200°C) and roll out one disc of dough to fit a 9-inch pie plate.

Pour the chicken filling into the pie crust, then roll out the second dough disc and place it over the filling.

Trim and crimp the edges to seal, cut a few slits in the top for venting, and bake for 35-45 minutes until golden brown and bubbly.