Rinse the chicken pieces and pat them dry with paper towels. Season generously with salt and pepper.

In a very large pot or Dutch oven, heat the olive oil or butter over medium-high heat. Sear the chicken pieces in batches until browned on all sides, about 3 to 4 minutes per side. Remove the chicken and set aside.

Add the chopped onion, carrots, and celery to the pot. Sauté the vegetables until the onion is translucent and soft, about 5 to 7 minutes, scraping up any browned bits from the bottom of the pot.

Return the chicken to the pot. Add the cold water, dried thyme, and bay leaves. The water should fully cover the chicken and vegetables. Bring the mixture to a boil, then immediately reduce the heat to a gentle simmer. Cover the pot partially and cook for 45 to 60 minutes, or until the chicken is falling-off-the-bone tender.

Carefully remove all the chicken pieces from the broth and place them on a platter to cool slightly. Leave the broth in the pot. Once the chicken is cool enough to handle, remove and discard the skin and bones. Shred or dice the chicken meat into bite-sized pieces. Set the shredded chicken aside.

Strain the broth through a fine-mesh sieve into a clean bowl, discarding the cooked vegetables, herbs, and any solids. Return the strained broth to the pot.

Bring the strained broth back to a low simmer. Taste the broth and adjust the seasoning with salt and pepper as needed.

If you desire a thicker broth, create a slurry by whisking the 1/2 cup flour with 1 cup of cold water or cold broth in a small bowl until smooth. Slowly whisk the slurry into the simmering broth. Simmer for 5 to 10 minutes until the broth thickens slightly.

Stir the heavy cream or milk into the broth if using, and maintain a rapid, rolling boil before beginning the dumplings.

While the broth is simmering, prepare the dumpling dough. In a large mixing bowl, whisk together the 2 cups of flour, baking powder, 1 teaspoon of salt, and 1/2 teaspoon of black pepper.

Add the cold butter cubes to the dry ingredients. Using a pastry blender, a fork, or your fingertips, cut the butter into the flour mixture until the mixture resembles coarse crumbs.

Pour in the cold milk all at once. Stir the mixture quickly until just combined. The dough will be shaggy, slightly sticky, and stiff, but do not overmix.

Ensure the broth is at a vigorous, rolling boil.

Drop the dumpling dough by tablespoonfuls directly into the boiling broth. Work quickly, leaving space between each dumpling. Do not overcrowd the pot; cook in batches if necessary.

Once all the dumplings are added, immediately reduce the heat to low, cover the pot tightly with a lid, and cook without lifting the lid for exactly 15 minutes. The steam is essential for cooking the dumplings through.

After 15 minutes, remove the lid. The dumplings should be fluffy and doubled in size. Gently fold the shredded chicken meat back into the broth. Stir carefully to avoid breaking the dumplings.

Ladle the Chicken and Dumplings into large bowls, ensuring each serving has several dumplings and plenty of chicken. Garnish generously with fresh chopped parsley before serving.