01 - In a large mixing bowl, whisk together the all-purpose flour and salt. Incorporate the cold, cubed unsalted butter using a pastry cutter or your fingertips until the mixture resembles coarse meal with pea-sized pieces. Gradually add the ice water, mixing until the dough just comes together. Divide the dough in half, shape each portion into a disk, wrap tightly in plastic, and refrigerate for a minimum of 30 minutes.
02 - In a large skillet, melt 2 tablespoons of unsalted butter over medium heat. Add the diced onions, carrots, and celery, sautéing until softened, which typically takes about 6 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.
03 - Sprinkle the 1/3 cup of all-purpose flour into the sautéed vegetables, stirring constantly for 2 minutes to cook out any raw flour taste. Gradually whisk in the chicken broth and whole milk (or half-and-half). Continue to cook, stirring frequently, until the sauce thickens, approximately 5–7 minutes.
04 - Add the cooked chicken, frozen peas, dried thyme, dried sage, and season with salt and freshly ground black pepper to taste. Simmer for 2 minutes, then remove the skillet from the heat. Allow the filling to cool slightly.
05 - Preheat your oven to 400°F.
06 - On a lightly floured surface, roll out one disk of chilled pie dough to fit a 9-inch pie dish. Gently press the dough into the dish. Pour the prepared chicken pot pie filling evenly into the crust-lined dish.
07 - Roll out the second disk of pie dough and carefully lay it over the filling. Trim any excess dough from the edges and crimp to seal the top and bottom crusts together. Create a few small slits in the center of the top crust to allow steam to escape during baking.
08 - In a small bowl, beat the remaining large egg with 1 tablespoon of water to create an egg wash. Lightly brush the top crust of the pie with this mixture.
09 - Bake the chicken pot pie for 40–45 minutes, or until the crust is golden brown and the filling is visibly bubbling. If the edges begin to brown too quickly, loosely cover them with aluminum foil.
10 - Remove the pie from the oven and allow it to rest for 10 minutes before slicing and serving. This resting period helps the filling set.