In a 1-gallon resealable bag, mix 1/2 cup flour, 1/2 teaspoon salt, and 1/2 teaspoon pepper. Add chicken (4-5 pounds), a few pieces at a time, and toss to coat; shake off excess.
In a 6-quart Dutch oven or large heavy-bottomed pot, heat oil (2 tablespoons) over medium-high heat. Brown chicken in batches on all sides; transfer to a plate. Add more oil as needed.
Add onion (1), carrots (2), and celery (2 ribs) to Dutch oven and cook, stirring often, being sure to scrape up all the brown bits on the bottom of the Dutch oven, until onion is tender about 6-8 minutes Add garlic (4 cloves); cook and stir 1 minute longer.
Sprinkle 1/4 cup flour over vegetables. Stir to evenly coat the vegetables. Gradually add stock (6 cups), stirring constantly. Stir in wine (1/2 cup), sugar (2 teaspoons), thyme sprigs (4), bay leaves (2), and peppercorns (5).
Return chicken to pan; bring to a boil. Reduce heat and simmer until the chicken is falling off the bones tender, about 45 minutes 1 hour. Remove the chicken from the pot and set aside until it's cool enough to handle. While the chicken is cooking, prepare the dumplings.
Make Dumplings: Whisk together flour (1 1/2 cups), baking powder (1 1/2 teaspoon), salt (3/4 teaspoon), and pepper (1/4 teaspoon) in a medium mixing bowl. In a small bowl, whisk together the buttermilk (3/4 cup) and egg (1) and add to the dry ingredients, followed by the melted butter (2 tablespoons). Using a wooden spoon stir just to combine. (Do not overmix or dumplings will be tough.); set aside.
Transfer the chicken from the pot to a plate. Discard thyme stems and bay leaves and skim fat from the soup. Remove skin and bones from chicken and discard. Using 2 forks, coarsely shred meat into 1- to 1-1/2-in. pieces; return to soup. Stir in heavy cream (1/2 cup). Cook, covered, on high until the mixture reaches a simmer (Not a boil).
Drop generous quarter-size dollops on top of simmering soup. Cover the pot, and reduce heat to low. Cook, undisturbed, until the dumplings are puffed and totally cooked through, about 18 to 22 minutes, or until a toothpick inserted in the center of the dumplings comes out clean. Sprinkle top with parsley (1 tablespoon) and thyme (2 teaspoons) and serve.