Heat 2 teaspoons of the oil in a large Dutch oven over medium-high heat until just smoking.

Add half of the chicken and cook until golden on both sides, about 10 minutes. Transfer the chicken to a plate and remove the browned skin.

Pour off the chicken fat and reserve.

Return the pot to medium-high heat and repeat with the remaining 2 teaspoons oil and the remaining chicken. Pour off and reserve any chicken fat.

Add the butter to the Dutch oven and melt over medium-high heat. Add the carrots, celery, onion, and 1/4 teaspoon salt and cook until softened, about 7 minutes.

Stir in the flour. Whisk in the sherry, scraping up any browned bits.

Stir in the broth, milk, thyme, and bay leaves.

Nestle the chicken, with any accumulated juices, into the pot. Cover and simmer until the chicken is fully cooked and tender, about 1 hour.

Transfer the chicken to a cutting board. Discard the bay leaves. Allow the sauce to settle for a few minutes, then skim the fat from the surface using a wide spoon.

Shred the chicken, discarding the bones, then return it to the stew.

Stir the warmed milk mixture into the flour mixture with a wooden spoon until incorporated and smooth.

Return the stew to a simmer, stir in the peas and parsley, and season with salt and pepper.

Drop golf-ball-sized dumplings over the top of the stew, about 1/4 inch apart (you should have about 18 dumplings). Reduce the heat to low, cover, and cook until the dumplings have doubled in size, 15 to 18 minutes. Serve.