01 - Combine flour and salt in a large bowl. Cut in cold butter using a pastry cutter or fingers until pea-sized lumps form. Gradually add ice water, mixing until dough forms. Divide dough in half, shape into disks, wrap in plastic, and chill at least 30 minutes.
02 - Melt butter in a large skillet over medium heat. Sauté onion, carrots, and celery until tender, about 6 minutes. Add garlic; cook 1 minute more.
03 - Sprinkle flour over vegetables, stirring constantly for 2 minutes. Gradually whisk in chicken broth and milk, cooking until thickened, 5 to 7 minutes.
04 - Stir in chicken, peas, thyme, sage, salt, and pepper. Simmer for 2 minutes. Remove from heat and let cool slightly.
05 - Set oven temperature to 400°F (200°C).
06 - Roll out one dough disk to fit a 9-inch pie dish. Press dough into the dish and pour in filling.
07 - Roll out second dough disk and lay over filling. Trim and crimp edges to seal. Cut small slits on top for steam release.
08 - Beat egg with 1 tablespoon water and brush over the crust surface.
09 - Bake 40 to 45 minutes until crust is golden and filling bubbles. Cover edges with foil if browning too quickly.
10 - Allow pot pie to rest for 10 minutes before slicing.