For the stew, pat the chicken dry with paper towels and season with salt and pepper. Heat the oil in a large Dutch oven over medium-high heat. When the oil is shimmering, add half of the chicken and cook until golden on both sides, about 10 minutes. Transfer to a plate and repeat with the remaining chicken. Set aside to cool. Add butter to the pan and melt over medium-high heat. Add onion and 1⁄2 teaspoon of salt and cook until soft, about 5 minutes. Add carrots, fennel, and parsnip, and cook until just tender, 4–5 minutes. Add garlic and when it releases its fragrance stir in flour and cook, stirring, for 2–3 minutes. Add sherry and stir, scraping up any bits stuck to the pan. Add stock and stir to combine. Tie thyme, bay leaves, and parsley together with kitchen twine, drop the bundle into the broth, and add the chicken, taking care to fully immerse it in the liquid. Bring stew to a boil, reduce heat to low, cover, and simmer until the chicken is fully cooked, about 1 hour. Remove from heat. Transfer the chicken to a cutting board. Remove the bundle of herbs and discard. When chicken is cool enough to handle, remove skin and bones, and discard. Tear the meat into bite-size pieces. Using a skimmer or wide spoon, skim the fat from the surface of the sauce in the pot. Return the chicken and any drippings to the stew. For the dumplings, bring the stew to a simmer. In a medium bowl, whisk together the flour, baking powder, baking soda, salt, and pepper. Add the butter, buttermilk, and herbs, and stir to form a thick, very sticky dough. Using your hands, roll the dough into balls about the size of a golf ball. Drop the dough balls into the stew, spaced about 1⁄4” apart. Cover and simmer over low heat until the dumplings are firm and cooked through, about 15 minutes.