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.
Add the butter to the same pot and melt over medium-high heat. Add the onion and 1/2 teaspoon of salt and cook until soft, about 5 minutes. Add the carrots, fennel and parsnip and cook until just tender, 4 to 5 minutes. Add the garlic and when it releases its fragrance, stir in the flour and cook, stirring, for 2 to 3 minutes. Add the sherry and stir, scraping up any bits stuck to the pot. Add the broth and stir to combine.
Tie the thyme, parsley and bay leaves together with kitchen twine, drop the bundle into the broth and add the chicken, taking care to fully immerse it in the liquid. Bring the stew to a boil, reduce the heat to low, cover and simmer until the chicken is fully cooked, about 1 hour.
Remove the pot from the heat and transfer the chicken to a cutting board. Remove the bundle of herbs and discard. When the chicken is cool enough to handle, remove the 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 pot and bring the stew to a simmer.
For the dumplings: In a medium bowl, whisk together the flour, baking powder, salt, baking soda and pepper. Add the buttermilk, butter, thyme, parsley and dill 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 inch apart. Cover and simmer over low heat until the dumplings are firm and cooked through, about 15 minutes.