For the crust: In a large bowl, whisk together flour and salt. Cut in cold butter with a pastry blender or your fingers until mixture resembles coarse crumbs. Gradually add ice water, one tablespoon at a time, mixing until dough just comes together. Divide dough in half, flatten into discs, wrap in plastic, and refrigerate for at least 30 minutes.
For the filling: In a large saucepan, melt butter over medium heat. Add onion and celery and cook until softened, about 5 minutes. Stir in flour and cook for 1 minute.
Gradually whisk in chicken broth and milk. Bring to a simmer, stirring constantly, until thickened. Stir in shredded chicken, peas, carrots, black pepper, and thyme. Cook for 5 minutes, stirring occasionally.
Preheat oven to 400°F (200°C). On a lightly floured surface, roll out one dough disc to fit the bottom of a 9-inch pie plate. Pour the filling into the crust. Roll out the second dough disc and place it over the filling. Crimp the edges to seal and cut slits in the top crust to allow steam to escape.
Bake for 30-35 minutes, or until crust is golden brown and filling is bubbly.
Let stand for 10 minutes before serving.
Notes
This chicken pot pie can be made ahead of time and reheated. For a quicker version, use store-bought pie crust.