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 immediately reduce the heat 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 cooked 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 (this should yield about 3 cups of shredded chicken).

Pour the liquid and vegetables remaining in the pot through a fine-mesh sieve or colander into a separate bowl. Discard the cooked vegetables. Measure out 6 cups of the reserved chicken stock and set aside. Wipe out the stockpot and return it to the stovetop.

Over medium heat, melt the 4 tablespoons of butter in the cleaned pot. Whisk in the 4 tablespoons of flour to create a roux. Cook the roux for 1 to 2 minutes, stirring constantly, until it forms a smooth, pale paste and smells slightly nutty.

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

Stir in the heavy cream, the shredded chicken, and the frozen peas (if using). Season the stew base with additional salt and pepper as needed. Keep the stew at a steady, 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 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.

Cut the dough into 1-inch squares or drops using a knife or small cutter.

Carefully drop the biscuit pieces onto the surface of the simmering stew, ensuring the stew is bubbling gently but not violently boiling. Leave 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. After 15 minutes, remove the lid. The dumplings should be doubled in size, firm to the touch, and cooked completely through. Serve immediately.