Adjust oven rack to low-center position; heat oven to 400 degrees. Put chicken and broth in a dutch oven over medium heat. Cover and bring to simmer, simmer until chicken is done. Transfer meat to a large bowl and broth to another bowl.
Increase heat to medium-high; heat oil in Dutch oven. Add onion, carrots, and celery and saute until just tender, about 5 minutes. Season to taste with salt and pepper. Meanwhile, shred chicken into bite-size pieces. Transfer cooked vegetables to bowl with chicken; set aside.
Heat butter over medium heat in Dutch oven. When foaming subsides, add flour; cook for about 1 minutes. Whisk in reserved chicken broth, milk, any accumulated chicken juices and thyme. Bring to a simmer, reduce heat to low and continue to simmer, stirring, until sauce fully thickens, about 1 minute. Season to taste with salt and pepper; stir in sherry.
Pour sauce over chicken-vegetable mixture; stir to combine. Stir in peas and parsley, and adjust seasonings. Pour mixture into a 13-by-9 inch pan or any shallow baking dish of similar size or into six 12-ounce ovenproof ramekins or bowls. Top as desired. Bake until pastry is golden brown and filling is bubbly, about 30 minutes for large pies, 20 to 25 minutes for smaller pies