01 - Whisk flour and salt in a large bowl. Cut in cold butter until pea-sized pieces form. Gradually add ice water, mixing until dough comes together. 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 diced onions, carrots, and celery until softened, about 6 minutes. Add garlic and cook for 1 more minute.
03 - Sprinkle flour into the skillet, stirring constantly for 2 minutes to eliminate raw flavor. Gradually whisk in chicken broth and milk. Cook, stirring, until the mixture thickens, approximately 5 to 7 minutes.
04 - Stir in chicken, peas, thyme, sage, salt, and pepper. Simmer for 2 minutes, then remove from heat and allow to cool slightly.
05 - Set oven temperature to 400°F (200°C).
06 - Roll out one dough disk to fit a 9-inch pie dish. Gently press the dough into the dish and pour in the filling.
07 - Roll out the second dough disk and place it over the filling. Trim excess dough and crimp edges to seal. Cut small slits in the top crust to allow steam ventilation.
08 - Beat the egg with 1 tablespoon water and brush the mixture evenly over the top crust.
09 - Bake for 40 to 45 minutes until the crust turns golden brown and filling bubbles. Cover crust edges with foil if browning excessively.
10 - Allow the pot pie to rest for 10 minutes before slicing and serving.