Place the chicken pieces, onion, carrots, celery, 2 teaspoons of salt, 1 teaspoon of pepper, thyme, and bay leaves into a large stockpot or Dutch oven. Pour the water or chicken stock over the ingredients, ensuring the chicken is covered.

Bring the liquid to a boil over high heat, then reduce the heat immediately to low, cover partially, and simmer gently for 45 to 60 minutes, or until the chicken is tender and cooked through.

Carefully remove all the chicken pieces from the pot and set them aside on a cutting board to cool slightly. Leave the vegetables and liquid in the pot. Once the chicken is cool enough to handle, discard the skin and bones. Shred or chop the meat into bite-sized pieces and set aside.

Pour the liquid and vegetables remaining in the pot through a fine-mesh sieve or colander into a separate bowl. Discard the cooked vegetables (they have given up most of their flavor).

Measure out 6 cups of the reserved chicken stock. Set aside the remaining stock in case you need to thin the stew later. Wipe out the stockpot or Dutch oven and return it to the stovetop over medium heat.

Add the 4 tablespoons of butter to the pot and allow it to melt. Whisk in the 4 tablespoons of all-purpose flour to create a roux. Cook the roux for 1 to 2 minutes, stirring constantly, until it forms a smooth paste and smells slightly nutty. Do not let it brown.

Slowly pour in the measured 6 cups of warm chicken stock, whisking continuously to prevent lumps. Bring the mixture to a simmer, stirring occasionally. Reduce the heat to low and let it simmer for 10 minutes, allowing the sauce to thicken slightly.

Stir in the heavy cream or milk and the shredded chicken. Season the stew base with additional salt and pepper as needed. Add the frozen peas (if using). Keep the stew at a gentle simmer.

In a large mixing bowl, whisk together the 2 cups of flour, baking powder, salt, and pepper for the dumplings.

Cut the cold butter cubes into the flour mixture using a pastry blender or your fingertips until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.

Pour in the cold buttermilk (or milk/lemon mixture) and mix just until a shaggy dough forms. Do not overmix.

Lightly dust a clean surface with flour. Turn the dough out and gently pat or roll it out to about 1/2-inch thickness. Using a small round cutter, or simply using a knife, cut the dough into 1-inch squares or drops.

Ensure the chicken stew is at a steady, gentle simmer (it should not be violently boiling). Carefully drop the biscuit pieces onto the surface of the simmering stew, leaving a small amount of space between each piece.

Immediately cover the pot tightly with a lid.

Allow the dumplings to steam and cook for 15 to 20 minutes without lifting the lid. The internal heat and steam are crucial for the biscuits to rise and cook thoroughly.

After 15 minutes, remove the lid. The dumplings should be doubled in size, firm to the touch, and cooked completely through. Serve the Chicken and Dumplings immediately, garnishing with fresh parsley if desired.