Using 2 large forks, shred breast, thigh, and drumstick meat, and any remaining meat from the chicken carcass into bite-size pieces. Transfer shredded meat to a bowl and discard bones and skin.

For the Soup: In now-empty pot or Dutch oven, melt butter over medium heat. Once melted, add onion, carrot, celery, thyme, and garlic. Cook, stirring occasionally, until vegetables soften, 7 to 8 minutes. Sprinkle with flour; cook, stirring constantly, until vegetables are evenly coated and flour begins to stick to the bottom of the pot, 1 to 2 minutes.

Whisk in the chicken stock, evaporated milk, salt, and Worcestershire sauce; bring to a simmer over medium heat. Cook, stirring often, until mixture thickens, about 5 minutes. Season to taste with salt and pepper as needed. Reduce heat to low to maintain a simmer while preparing the dumplings.

For the Dumplings: In a medium bowl, whisk together the flour, baking powder, salt, baking soda, pepper, and thyme. Using a flexible spatula, incorporate sour cream, butter, and milk, stirring until fully combined. (Avoid overmixing, which will result in tough dumplings.)

Stir the shredded chicken into the soup. Using 2 large spoons, work quickly to gently drop golf ball–size portions of dumpling dough into simmering soup; you should have about 15 dumplings. Some dumplings may dissolve slightly into the soup. This is OK as it will further thicken the soup. Cover and gently simmer until dumplings are fluffy, cooked through, and have doubled in size, about 10 minutes. Top with parsley and serve.