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Heat oven to 425 degrees. Line a sheet pan with parchment paper and set aside.
Season chicken with salt and pepper. Heat oil in a 10-inch cast-iron skillet (or other heavy bottom skillet) over medium heat. Sear chicken, skin-side down, until deeply golden brown and some of the fat has begun to render out, 5 to 8 minutes. Rotate chicken and continue to cook on all sides until evenly browned and cooked through, an additional 8 to 10 minutes. Remove chicken, keeping all the drippings in the skillet, and let cool slightly on a plate.
To the drippings, add carrots, celery, garlic, onion and thyme. Season with salt and pepper and cook, stirring occasionally until vegetables are tender and just cooked through, about 5 to 8 minutes. Add flour and stir to coat all the vegetables. Cook, stirring constantly, until the flour has started to turn a light golden brown with no white, floury bits left, about 3 minutes.
Slowly add the chicken broth or stock ½ cup at a time, using a wooden spoon to blend and scrape up any browned bits on the bottom of the skillet. Once all the broth or stock is added, bring to a simmer and remove from heat.
Remove the bones and skin from the chicken and shred the meat and skin (discard any skin that is too thick or rubbery) into bite-size pieces or, alternatively, cut it with a knife and fork into bite-size pieces. (If you discover at this stage any pieces of chicken that are not completely cooked through, don’t worry; they will finish cooking in the oven.) Add the peas, parsley and chicken and season once more with salt and pepper.
Carefully unfold the puff pastry and working quickly, place it on top of the skillet (do not worry about it being a perfect circle), letting some of the dough hang off the sides (this will keep it from shrinking too much).
Whisk the egg with 1 teaspoon of water and brush onto the top of the puff pastry. Cut three 2½-inch slits, about 1-inch apart and place the skillet on top of the sheet pan to catch any drips in the oven.
Bake until puff pastry is golden brown on top and filling is starting to bubble up, 20 to 25 minutes. Reduce temperature to 375 degrees and continue to cook until puff pastry is baked through all the way and filling has thickened considerably, another 15 to 20 minutes.
Remove from heat and let cool slightly before eating.