1. Roll dough between 2 sheets of parchment paper into 11-inch circle. Remove top parchment sheet. Fold in outer 1/2 inch of dough to make 10-inch circle. Using index finger of one hand and thumb and index finger of other hand, crimp edge of dough to make attractive fluted rim. Using paring knife, cut 4 oval-shaped vents, each about 2 inches long and 1/2 inch wide, in center of dough. Transfer dough, still on parchment, to baking sheet and refrigerate until firm, about 15 minutes.
2. Adjust oven rack to middle position and heat oven to 400 degrees. Brush dough with egg mixture and bake until golden brown, 10 to 12 minutes, rotating sheet halfway through baking. Transfer crust, still on sheet, to wire rack and let cool; do not turn off oven.
3. Heat 10-inch cast-iron skillet over medium heat for 3 minutes. Melt butter in skillet. Add carrots, celery, onion, 1/4 teaspoon salt, and 1/4 teaspoon pepper and cook until softened and lightly browned, 5 to 7 minutes. Stir in thyme and cook until fragrant, about 30 seconds. Stir in flour and cook for 2 minutes. Slowly whisk in broth, scraping up any browned bits and smoothing out any lumps, and bring to simmer.
4. Pound thicker ends of chicken breasts as needed to create even thickness. Nestle chicken into skillet. Reduce heat to gentle simmer, cover, and cook until chicken registers 160 degrees and sauce has thickened, 10 to 15 minutes, flipping chicken halfway through.
5. Transfer chicken to carving board, let cool slightly, then shred into bite-size pieces using 2 forks. Stir shredded chicken, peas, cream, parsley, and sherry into skillet. Season with salt and pepper to taste.
6. Place parbaked pie crust on top of filling, transfer skillet to oven, and bake until crust is deep golden brown and filling is bubbling, about 10 minutes. Let pot pie cool for 10 minutes before serving.