Step 1: Place the chicken, chicken broth, bay leaf, and peppercorns (if using) in a large Dutch oven or pot. Bring to a boil, then reduce heat, cover, and simmer for 20-25 minutes, or until the chicken is cooked through and tender.
Step 2: Carefully remove the cooked chicken from the pot and set aside to cool slightly. Discard the bay leaf and peppercorns. Strain the broth through a fine-mesh sieve into a clean bowl; set the broth aside. Shred the cooled chicken using two forks or your hands.
Step 3: While the chicken cools, prepare the dumpling dough. In a large bowl, combine the self-rising flour (or homemade equivalent) and cold milk. Mix with a spoon or your hands until a soft, shaggy dough forms. Do not overmix.
Step 4: Lightly flour a clean surface. Turn the dough out and gently knead for 1-2 minutes until it comes together. Divide the dough in half.
Step 5: Roll out one half of the dough very thinly, about 1/8-inch thick, into a large rectangle. Using a pizza cutter or sharp knife, cut the dough into approximately 1x2-inch strips or squares. Repeat with the second half of the dough.
Step 6: Return the strained chicken broth to the Dutch oven and bring it to a gentle boil over medium-high heat. Reduce the heat to a simmer. Drop the dumpling strips into the simmering broth, a few at a time, ensuring they don't stick together. Cook for 8-10 minutes, stirring occasionally, until the dumplings are tender and cooked through, and the broth has slightly thickened.
Step 7: Stir the shredded chicken back into the pot with the dumplings and broth. Add the heavy cream and butter, stirring until the butter is melted and everything is well combined and warmed through.
Step 8: Taste and season generously with salt and freshly ground black pepper. Serve hot, garnished with fresh parsley if desired.