- Pull the meat from the chicken and tear it into large bite-sized pieces, reserving the bones and skin; cover and refrigerate until needed.
- Place the bones and skin in a large saucepan or small pot. Add the stock and cold water. Bring to a boil, reduce the heat, and simmer until the carcass falls apart and the liquid reduces to about 6 cups, about 45 minutes. Strain the broth into a clean bowl and discard the solids.
- Heat the butter in a Dutch oven over medium heat. Add the onion, celery, carrots, thyme, and a pinch of salt and stir to coat. Cook, stirring often, until the vegetables begin to soften, about 8 minutes. Add the broth and simmer until the vegetables are tender, about 10 minutes.
- Season with salt and pepper. Stir in the chicken and keep warm overlow heat.
- To make the dumpling dough, place the flour in a medium bowl. Use a pastry blender or your fingertips to work in the butter and shortening until the mixture is crumbly and flecked with flakes of fat. When pressed against the back of a thumb, a bit of the mixture should cling like a small leaf. Slowly stir in the half-and-half. The dough should be soft and sticky but firm enough to hold together on a spoon.
- Bring the chicken stew to a low boil over medium-high heat. Use a 1-ounce scoop or two spoons to drop golf-ball-size dumplings evenly over the surface of the stew. Cover the pot and cook until the dumplings are firm, fluffy, and fairly dry on top, about 20 minutes.
- Serve at once, sprinkled with a little parsley for color.