In a bowl, combine 1/2 cup of all-purpose flour, salt, and freshly ground pepper. Mix well to season the flour.
Take the cut-up chicken and coat each piece in the seasoned flour mixture. Shake off excess flour.
In a Dutch oven over medium heat, add 2 tablespoons of oil. Brown the floured chicken pieces in the Dutch oven. Be cautious not to cook them entirely, just brown the exterior for flavor. Once browned, remove the chicken from the Dutch oven and set aside.
In the same Dutch oven, sauté the chopped onion, carrots, and celery until softened. Add the minced garlic and cook for about 1 minute, until fragrant.
Sprinkle the remaining 1/4 cup flour over the vegetables and stir well. Gradually pour in the chicken stock, stirring to avoid lumps and loosen the browned bits. Add the apple cider, sugar, bay leaves, and whole peppercorns. Bring to a simmer.
Return the browned chicken to the Dutch oven. Cover and simmer for about 25 minutes, or until the thickest pieces reach at least 165°F away from the bone and are tender enough to shred. Larger pieces may take longer. Transfer the cooked chicken to a clean plate to cool slightly. Remove and discard the bay leaves and whole peppercorns. Skim excess fat from the broth if needed.
In a separate bowl, combine 1-1/3 cups of all-purpose flour, baking powder, and salt. In another bowl, mix together milk and melted butter. Pour the wet ingredients into the dry ingredients and stir until just combined.
Shred the cooked chicken using two forks, discarding the bones and skin. Return the shredded chicken to the pot and bring the soup back to a gentle simmer.
Drop small, evenly sized spoonfuls of dough onto the gently simmering soup, leaving room for them to expand. Cover and cook for about 15 minutes without lifting the lid. A toothpick inserted into the center of a dumpling should come out clean, with no wet dough. Cover and cook a little longer if needed.
Once the dumplings are cooked, lower the heat and gently stir in the heavy whipping cream, parsley, and thyme without breaking up the dumplings. Heat through and season with additional salt and pepper to taste.
Ladle the Chicken and Dumplings into bowls. Garnish with additional herbs if desired. Serve hot and enjoy your comforting and flavorful Chicken and Dumplings!