Cook the Chicken:
In a large pot or Dutch oven, heat the olive oil or butter over medium heat. Add the chicken breasts or thighs and cook until browned on both sides, about 6-7 minutes per side. Once browned, remove the chicken from the pot and set aside.
Sauté the Vegetables:
In the same pot, add the diced onion, carrots, and celery. Cook for about 5 minutes, until softened. Add the garlic and cook for another minute until fragrant.
Make the Broth:
Pour in the chicken broth, scraping up any browned bits from the bottom of the pot. Stir in the thyme, salt, and pepper. Bring the mixture to a simmer.
Shred the Chicken:
While the soup is simmering, shred the cooked chicken into bite-sized pieces. Return the chicken to the pot.
Add the Cream:
Stir in the heavy cream or half-and-half and continue simmering for about 5 minutes, allowing the soup to thicken slightly.
Make the Dumplings:
In a mixing bowl, combine the flour, baking powder, salt, garlic powder, and pepper. Cut in the cold, cubed butter using a pastry cutter or your hands until the mixture resembles coarse crumbs. Add the milk and fresh parsley, stirring until just combined. Be careful not to overmix.
Drop the Dumplings:
Scoop spoonfuls of the dumpling batter and carefully drop them onto the simmering soup. Try to evenly space them out. Cover the pot with a lid and let the dumplings steam for 15-20 minutes, or until they’ve puffed up and are cooked through.
Serve and Garnish:
Once the dumplings are cooked, ladle the soup and dumplings into bowls. Garnish with fresh parsley and serve hot.