In a large bowl, whisk flour and salt. Add cold butter cubes and toss to coat. Using a pastry cutter or your fingertips, work butter into flour until mixture resembles coarse meal with some pea-sized pieces. Drizzle ice water over mixture 1 tablespoon at a time, stirring with a fork until dough begins to form clumps. Gather into a ball, divide in half, flatten into disks, wrap in plastic, and refrigerate for at least 30 minutes.

Preheat oven to 400°F (200°C). In a large skillet over medium-high heat, melt 1 tablespoon butter. Season chicken pieces with a pinch of salt and pepper, then sear until golden and cooked through, reaching an internal temperature of 165°F (74°C), about 5-6 minutes. Remove to a plate. In the same skillet, add remaining 2 tablespoons butter, onion, carrots, and celery. Cook until softened, about 5-7 minutes.

Sprinkle 1/3 cup flour over the softened vegetables and stir constantly for 1-2 minutes to cook out the raw taste. Slowly pour in chicken broth while whisking to prevent lumps. Add heavy cream and bring to a simmer. Cook until thickened, about 3-4 minutes. Stir in cooked chicken, frozen peas, thyme, 1 tsp salt, and pepper. Remove from heat and let cool slightly.

On a lightly floured surface, roll one dough disk into a 12-inch circle. Ease it into a 9-inch deep-dish pie pan, leaving a 1-inch overhang. Pour in the cooled filling. Roll out the second disk and place over filling. Trim excess dough, fold overhang over the top crust, and crimp edges decoratively. Cut several slits in the top crust to vent steam. Brush with beaten egg for a glossy golden finish.

Place pie on a baking sheet to catch any drips. Bake at 400°F for 20 minutes, then reduce heat to 350°F (175°C) and continue baking for 25-30 minutes, or until crust is deep golden and filling is bubbly. If edges brown too quickly, cover with foil. Let rest at least 15 minutes before slicing.