Create the Flavorful Broth: Place the whole chicken, roughly chopped carrots, roughly chopped celery, quartered onion, bay leaves, peppercorns, and thyme sprigs into a large stockpot. Pour in the chicken broth and enough water to cover the chicken completely. Bring to a gentle simmer over medium-high heat, then reduce the heat to low, cover, and simmer for 1.5 to 2 hours, until the chicken is fall-off-the-bone tender. Skim any foam that rises to the surface.

Shred the Chicken: Carefully remove the chicken from the pot and set it on a cutting board to cool slightly. Once cool enough to handle, pull all the meat from the bones, discarding the skin and bones. Shred the meat into bite-sized pieces and set aside. Strain the broth through a fine-mesh sieve into a large bowl, discarding the cooked vegetables and aromatics.

Build the Soup Base: Wipe out the stockpot and place it back on the stove over medium heat. Melt the 4 tablespoons of butter. Add the finely diced onion, carrots, and celery and cook, stirring occasionally, until softened, about 8-10 minutes.

Thicken the Soup: Sprinkle the 1/2 cup of all-purpose flour over the cooked vegetables and stir constantly for one minute to cook out the raw flour taste. Slowly whisk in the reserved chicken broth, a cup at a time, ensuring each addition is smooth before adding the next. Bring the mixture to a simmer and cook for 5 minutes, until it has thickened into a gravy-like consistency.

Prepare the Dumpling Dough: While the soup simmers, whisk together the 2 cups of flour, baking powder, 1 tsp salt, and 1/2 tsp pepper in a medium bowl. Pour in the cold buttermilk and stir with a fork just until a shaggy, soft dough forms. Do not overmix.

Cook the Dumplings: Bring the soup to a steady, gentle simmer (not a rolling boil). Drop spoonfuls of the dumpling dough onto the surface of the simmering soup. Once all the dough is in the pot, reduce the heat to low, cover the pot tightly, and let the dumplings steam for 15 minutes. Do not lift the lid.

Finish and Serve: After 15 minutes, remove the lid. The dumplings should be puffed and cooked through (a toothpick inserted into the center should come out clean). Gently stir the shredded chicken and fresh parsley into the soup. Taste and adjust seasoning with salt and pepper as needed before serving hot.