For the stew: Pat chicken dry with paper towels and season with salt and pepper. Heat oil in a large Dutch oven over medium-high heat until just smoking. Brown chicken well, 10 to 12 minutes; transfer to plate and remove skin.
Add carrots, onions, celery, and 1/8 teaspoon salt to fat left in pot, cover, and cook over medium-low heat, stirring occasionally, until softened, 8 to 10 minutes. Stir in sherry, scraping up any browned bits. Add broth, thyme, and chicken and any accumulated juices. Bring to simmer, cover, and cook until chicken registers 160 degrees F, about 20 minutes.
Remove pot from heat; transfer chicken to plate. When cool enough to handle, shred meat into large pieces, discarding bones.
For the dumplings: Whisk flour, sugar, salt, and baking soda together in a large bowl. In separate bowl, stir chilled buttermilk and melted butter together until butter forms small clumps, then whisk in egg white. Add buttermilk mixture to flour mixture and stir with rubber spatula until just incorporated.
Return stew to simmer and stir in shredded chicken, peas, and parsley. Season with salt and pepper to taste. Using greased tablespoon measure, scoop and drop dumplings on top of stew about 1 inch apart. Wrap lid of Dutch oven with clean kitchen towel (keeping towel away from heat source) and cover pot (this will prevent steam droplets from dripping onto the dumplings and making them soggy). Cook over low heat until dumplings have doubled in size, 13 to 16 minutes. Serve.