Place 1/3 cup flour, salt, and pepper in a gallon ziploc bag. Seal bag and shake to mix. Open bag and add chicken. Seal bag and shake again to coat all of the chicken pieces.
Add 2 tablespoons oil to a hot 6-quart dutch oven. Add chicken and cook for 5 to 7 minutes, or until chicken is browned and cooked. Remove chicken to a bowl and set aside.
Add 1 tablespoon oil to the dutch oven along. Once oil is hot, add onion, celery, carrots, and garlic. Cook for 5 minutes, stirring often until veggies have softened.
Add 1/4 cup flour to the pot and stir to coat the veggies. Add chicken back to pot and pour in chicken broth. Stir everything together.
Bring soup to a boil. Once boiling, reduce heat to low, cover the pot, and simmer for 12 to 15 minutes. Stir occasionally to prevent chicken or veggies from sticking to the bottom of the pot.
While to soup cooks, mix up your dumpling dough. In a medium mixing bowl whisk together flour, baking powder, salt, and pepper.
Pour in butter and whisk. Pour in milk and whisk until a thick, sticky dough forms.
Uncover the soup and stir in the heavy cream.
Use a large spoon or medium-sized cookie scoop to portion the dumpling dough. Place the dumplings right on top of the soup. (Don't worry they'll float.) Space them around the pot as they will puff up while cooking. Cover the pot and cook for 15 minutes without removing the lid.
Uncover the pot and remove from heat. Sprinkle with parsley. Portion soup into bowls and top with 2 dumplings. Serve warm.
Check to see if your dumplings are done after 15 minutes. Remove one from the pot and cut open to make sure they are not still raw inside. If they are, recover the pot and cook for another 5 minutes.