Melt 1 tablespoon of butter in a large pot over medium heat. Add the chicken pieces and cook until lightly browned on all sides.
Add the onion, carrots, celery, and garlic to the pot. Cook for 5 minutes until the vegetables begin to soften.
Pour in the chicken broth and stir in the thyme, salt, and black pepper. Bring the mixture to a simmer and cook for 15 minutes.
While the broth simmers, mix the flour, baking powder, and remaining 1/2 teaspoon salt in a bowl. Stir in the milk and remaining 2 tablespoons of melted butter until a soft dough forms.
Drop spoonfuls of the dough into the simmering broth. Cover the pot and cook for 12 minutes without lifting the lid so the dumplings steam properly.
Remove the pot from the heat and sprinkle the chopped parsley over the top before serving.