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Adjust oven rack to the lower-middle position and heat the oven to 400 degrees. Put the chicken and broth in a small Duch oven or stockpot over medium heat. Cover, bring to a simmer, and simmer until the chicken is just done, 8 to 10 minutes. Transfer the chicken to a large bowl, reserving the broth in a measuring cup. Increase the heat to medium-high and heat the oil in the now-empty pan. Add the onion, carrots, and celery and saute until just tender, about 5 minutes. Season with salt and pepper to taste. While the vegetables are sauteing, shred the meat into bite-size pieces. Transfer the cooked vegetables to a bowl with the chicken: set aside. Heat the butter over medium heat in the again-empty pan. When the foaming subsides, add the flour; cook about 1 minute. Whisk in the reserved chicken broth, the milk, any accumulated chicken juices, and the thyme. Bring to a simmer, then continue to simmer until the sauce fully thickens, about 1 minute. Season with salt and pepper to taste and stir in the sherry. Pour the sauce over the chicken mixture and stir to combine. Stir in the peas and parsley. Adjust the seasonings. (The mixture can be covered and refrigerated overnight; reheat before topping with the pastry.) Pour the mixture into a 13 by 9-inch baking pan (or shallow baking dish of similar size) or six 12- ounce ovenproof dishes. Top with the desired pastry. Bake until the pastry is golden brown and the filling is bubbling, 30 minutes for the large pot pie and 20 to 25 minutes for the smaller pies. Serve hot.
