Adjust oven rack to low-center position; heat oven to 400°F. Put chicken and broth  in a Dutch oven over medium heat. Cover and bring to simmer; simmer until chicken is just done, 8 to 10 minutes. Transfer meat to a large bowl; transfer both to another bowl.

Increase heat to medium-high; heat oil in Dutch oven. Add onion, carrots, and  celery and sauté 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 in Dutch oven over medium heat. When foaming subsides, add flour;  cook for about 1 minute. 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 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. Let rest for 10 minutes and serve hot.