Pat thighs dry and season with salt and pepper. Heat oil in large pot until shimmering. Add chicken thighs, skin side down. Cook until skin is crisp and well browned, 5-7 min., flip and cook second side 5-7 min. longer. Transfer to plate and discard all but 1 tsp fat.
Add onions, carrots, and celery to pot, cook stirring occasionally, until caramelized, 7-9 min. Stir in sherry, scraping up any browned bits. Stir in broth and thyme. Return chicken thighs and accumulated juices to pot and add chicken wings. Bring to simmer, cover, cook until thigh meat offers no resistance when poked with tip of knife but still clings to bones, about 45 minutes. Remove thighs from pot and simmer for additional 15 min.
Remove pot from heat and allow broth to settle 5 min., then skim fat from surface using wide spoon. Remove meat from thighs and wings and return to pot. Stew can be refrigerated and reheated for later use.
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 wet 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, scoop level amount of batter and drop over top of stew. Continue for remainder of batter (about 24 dumplings). Wrap lid of pot with clean kitchen towel and cover. Simmer gently until dumplings have doubled in size and toothpick comes out clean, 13-16 min. Serve in bowls immediately.