Chicken and Vegetables:
Heat the chicken stock to a gentle simmer in a medium pot.
In a separate, large pot, heat the butter or olive oil over medium-high heat. Pat dry the chicken pieces and sprinkle with salt. Working in batches, brown the chicken pieces, placing the pieces skin-side down first; this will render out fat you will use to build the stew later.
Once the chicken pieces are browned on both sides, remove them from the large pot, and turn off the heat. Remove and discard the skin from the chicken pieces and put the chicken pieces into the pot of simmering stock. Poach the chicken in the stock until cooked through, about 20 minutes or so. Remove the chicken pieces and set on a tray to cool for a few minutes. When the chicken pieces are cool to touch, pull the meat off the bones and cut into 2-inch chunks. Set aside.
Return the heat on the large pot to medium-high. When it is hot, add the onion, celery, carrot and thyme and sauté until soft, but not browned, about 4-5 minutes. Add the flour and mix well. The flour will absorb all the fat in the pot and will stick a little to the bottom. Turn the heat to medium-low and stir this constantly for 2-3 minutes. Do not let it burn.
Get a ladle ready and have the pot of simmering chicken stock nearby. Add the sherry to the flour vegetable mixture. It will sputter and seize up. Add a ladle of hot chicken stock to and stir well. It will be goopy. Add another ladle, then another, stirring all the while, until the broth comes together. Add the rest of the chicken stock, the reserved chicken meat. Increase the heat and bring to a simmer, then reduce the heat to maintain a gentle simmer while you make the dumplings.
Make the dumpling batter by sifting together flour, baking powder, and salt in a medium bowl. Add (optional) chopped fresh herbs. Add melted butter and milk to the dry ingredients. Gently mix with a wooden spoon until mixture just comes together. (Note: do not overmix! or your dumplings will turn out too dense.)
Drop dumpling batter into the simmering stew by heaping teaspoonfuls, over the surface of the stew. (Note that the dumplings will easily double in size as they cook.) Cover and simmer until dumplings are cooked through, about 15 minutes. Once you have covered the pan, do not uncover and peek while the dumplings are cooking! In order for the dumplings to be light and fluffy, they must steam, not boil. Uncovering the pan releases the steam. If after 15 minutes they are still not cooked through (use a toothpick or skewer to test) cover pan again, and cook for another 5 to 10 minutes.
Gently stir in peas, parsley and cream, if using. Add more salt to taste. Ladle portions of meat, sauce, vegetables, and dumplings into soup plates and serve. Note that the stew will continue to thicken the more it sits.