Adjust oven rack to low-center position; heat oven to 400 degrees. If using whole rotisserie chickens, separate meat from skin and bones, and tear into bite-size pieces. Discard skin and bones. Then remove pie dough from its box and follow directions for bringing to room temperature.
Heat oil in Dutch oven over medium-high heat. Add onions and celery; saute until just tender, about 5 minutes. Transfer to a large bowl along with the chicken; set aside.
Meanwhile, microwave chicken broth and milk in a microwave-safe bowl until steamy, 3 to 4 minutes. Heat butter over medium heat in the empty pan. When foaming subsides, whisk in flour and thyme; cook until golden, about 1 minute. Whisk in hot milk mixture. Bring to simmer, then continue to simmer until sauce fully thickens, about1 minute. Turn off heat, stir in sherry or wine, and season to taste with salt and pepper.
Stir chicken mixture, parsley and the peas and carrots into the sauce. Divide mixture between two 9-inch deep-dish pie plates. Top each with pie dough, and flute crust by pinching with your fingers. Set on a baking sheet and bake until pastry is golden brown and filling is bubbly, 30 to 35 minutes. (Optional: Wrap second potpie in freezer wrap and freeze for a later meal. Remove wrapping, place potpie on a baking sheet and bake at 400 degrees until golden and bubbly, about 1 hour.)