In a large pot or Dutch oven, melt butter over medium heat. Add chopped onion, carrots, and celery. Cook until softened, about 8-10 minutes.

Add chicken broth and chicken pieces to the pot. Bring to a boil, then reduce heat and simmer for 15-20 minutes, or until chicken is cooked through.

Remove chicken from the pot and shred or dice it. Return the shredded chicken to the pot.

In a small bowl, whisk together flour and heavy cream until smooth. Stir this mixture into the pot to thicken the broth.

Season with salt and pepper to taste.

In a medium bowl, whisk together flour, baking powder, and salt for the dumplings.

In a separate small bowl, whisk together milk and melted butter.

Pour the wet ingredients into the dry ingredients and stir until just combined. Do not overmix.

Drop spoonfuls of dumpling batter onto the simmering chicken mixture. Cover the pot and simmer for 15-20 minutes, or until dumplings are cooked through and fluffy. Do not lift the lid during this time.

Garnish with fresh parsley before serving.