Using a sharp knife, cut chicken breasts from breastbone and detach legs from carcass. In a pot, combine the detached chicken, celery, carrots, black peppercorns, bay leaves, salt, and water. Bring to a simmer over medium-high heat and cook uncovered until chicken breasts register 160℉ (71ºC), about 35 minutes.

Remove chicken breasts and transfer to a cutting board. Continue simmering the chicken legs until they reach 175℉ (80ºC), another 15 minutes. Remove thighs and drumsticks, and let broth simmer for another 25 minutes to develop flavor.

Remove chicken carcass and strain broth into a large bowl. Measure out 6 cups and set aside, reserving remaining broth for another use.

Shred the chicken meat into bite-size pieces and discard the bones and skin.

In the empty pot, melt butter over medium heat. Add onion, carrot, celery, thyme, and garlic, and cook until vegetables soften, about 7 to 8 minutes. Sprinkle with flour and cook until the flour begins to stick to the pot, about 1 to 2 minutes.

Whisk in the chicken stock, evaporated milk, salt, and Worcestershire sauce; bring to a simmer. Cook until the mixture thickens, about 5 minutes. Season to taste and reduce heat to maintain a simmer.

In a medium bowl, whisk together the flour, baking powder, salt, baking soda, pepper, and thyme. Incorporate sour cream, butter, and milk until fully combined.

Stir the shredded chicken into the soup. Drop portions of dumpling dough into simmering soup; cover and simmer until dumplings are fluffy and cooked through, about 10 minutes. Top with parsley and serve.