In a large pot or Dutch oven, melt 2 tablespoons butter over medium heat and sauté onion, carrots, and celery until softened, about 5 minutes.
Add garlic and cook for 1 minute, then pour in chicken broth and bring to a boil.
Add cooked chicken, thyme, and black pepper, then reduce heat to maintain a gentle simmer.
In a medium bowl, whisk together flour, baking powder, and salt, then stir in milk and melted butter until just combined to form a thick batter.
Drop dumpling batter by heaping tablespoonfuls onto the simmering soup, spacing them evenly.
Cover the pot with a lid and simmer for 15-20 minutes without lifting the lid, until dumplings are cooked through and fluffy.
Gently stir in fresh parsley and serve hot in bowls.