Adjust oven rack to middle position and heat oven to 400 degrees. Melt butter in 12‑inch oven safe skillet over medium heat. Add carrots, celery, onion, 1/4 teaspoon salt, and 1/4 teaspoon pepper and cook until vegetables are softened, about 5 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.
Nestle chicken into skillet. Reduce heat to low to maintain gentle simmer, cover, and cook until chicken registers 160 degrees and sauce has thickened, 15 to 20 minutes, flipping chicken halfway through cooking.
Transfer chicken to cutting board, let cool slightly, then shred into bite-size pieces. Stir shredded chicken, peas, cream, parsley, and sherry into skillet. Season with salt and pepper to taste. (Filling can be refrigerated for up to 1 day; return to simmer and add broth to loosen sauce if necessary.)
On clean, lightly floured counter, roll puff pastry into 12‑inch square. Loosely roll dough around rolling pin, then gently unroll it over filling in skillet, letting excess dough hang over edge. Cut four 2‑inch oval-shaped vents in center of dough, then brush dough with egg mixture. Bake until crust is golden brown and filling is bubbling, 30 to 35 minutes, rotating skillet halfway through baking.
Remove skillet from oven (skillet handle will be hot). Let cool for 10 minutes before serving.