FOR THE STEW Pat chicken thighs dry with paper towels and season with 1 teaspoon salt and 1/4 teaspoon pepper. Heat oil in large Dutch oven over medium-high heat until shimmering. Add chicken thighs, skin-side down, and cook until skin is crisp and well browned, 5 to 7 minutes. Using tongs, turn chicken pieces and brown on second side, 5 to 7 minutes longer; transfer to large plate. Discard all but 1 teaspoon fat from pot.
Add onions, carrots, and celery to now-empty pot; cook, stirring occasionally, until caramelized, 7 to 9 minutes. Stir in sherry, scraping up any browned bits. Stir in broth and thyme. Return chicken thighs, with any accumulated juices, to pot and add chicken wings. Bring to simmer, cover, and cook until thigh meat offers no resistance when poked with tip of paring knife but still clings to bones, 45 to 55 minutes.
Remove pot from heat and transfer chicken to cutting board. Allow broth to settle 5 minutes, then skim fat from surface using wide spoon or ladle. When cool enough to handle, remove and discard skin from chicken. Using fingers or fork, pull meat from chicken thighs (and wings, if desired) and cut into 1-inch pieces. Return meat to pot.
FOR THE DUMPLINGS Whisk flour, baking soda, sugar, and salt in large bowl. Combine buttermilk and melted butter in medium bowl, stirring until butter forms small clumps; whisk in egg white. Add buttermilk mixture to dry ingredients and stir with rubber spatula until just incorporated and batter pulls away from sides of bowl.
Return stew to simmer; stir in parsley and season with salt and pepper to taste. Using greased tablespoon measure (or #60 portion scoop), scoop level amount of batter and drop over top of stew, spacing about 1/4 inch apart (you should have about 24 dumplings). Wrap lid of Dutch oven with clean kitchen towel (keeping towel away from heat source) and cover pot. Simmer gently until dumplings have doubled in size and toothpick inserted into center comes out clean, 13 to 16 minutes. Serve immediately.