Make the Dough: In a food processor, pulse flour, salt, and sugar. Add cold butter and pulse until mixture resembles coarse meal. Drizzle in ice water, 1 tbsp at a time, pulsing until dough just begins to clump. Turn out, divide in two, shape into discs, wrap, and refrigerate for at least 1 hour.

Pre-bake Bottom Crust: On a floured surface, roll one dough disc to a 12-inch circle. Fit into a 9-inch deep-dish pie plate, trim edges, and prick the bottom with a fork. Line with parchment and pie weights or dried beans. Blind bake at 400°F (200°C) for 15-20 minutes until set. Remove weights and parchment. Let cool slightly.

Brown the Chicken: Season chicken generously. In a large Dutch oven, heat 1 tablespoon butter over medium-high heat. Brown chicken in batches without crowding. Transfer to a plate.

Build the Sauce Base: In the same pot, melt remaining 2 tablespoons butter. Sauté onion, carrots, and celery until softened. Add mushrooms and cook until browned. Add garlic and cook for 1 minute. Sprinkle flour over vegetables and cook, stirring constantly, for 2 minutes to form a golden roux.

Finish the Filling: Gradually whisk in chicken stock, then milk and cream. Bring to a simmer, stirring until thickened. Stir in sherry, thyme, peas, and the browned chicken (with any juices). Season heavily with salt and pepper. The filling should be very thick. Let cool for 15-20 minutes.

Assemble & Bake: Roll the second dough disc to an 11-inch circle. Fill the pre-baked bottom crust with the cooled filling. Brush the rim with egg wash. Place the top crust over the filling, trim, and crimp the edges to seal. Cut several steam vents in the top. Brush the entire top with egg wash. Chill assembled pie for 20 minutes. Preheat oven to 400°F (200°C).

Bake: Place pie on a baking sheet and bake for 45-55 minutes, until the crust is deep golden brown and the filling is actively bubbling through the vents. Let the pie rest for at least 20 minutes before serving to allow the filling to set.