Season the chicken pieces generously with salt and pepper. Heat the olive oil in a large Dutch oven over medium-high heat. Sear the chicken pieces for about 6 to 8 minutes, turning as needed to get some golden-brown spots. The chicken should still be raw in the center. Do not cook the chicken completely.
Remove the chicken from the pot and set aside to cool slightly, about 10 minutes. Once it’s cool enough to handle, remove the bones and cut the chicken into cubes. Set aside.
Add the butter to the Dutch oven and let it melt. Stir the melted butter around the pot to loosen the browned bits from the chicken, and combine with them. This step adds tons of flavor to the soup.
Add the finely diced onion, carrots, and celery. Cook for 5 to 7 minutes until they’re soft and aromatic. Add the garlic and cook for another minute.
Sprinkle the flour over the vegetables and stir to coat. Cook for 2 minutes with stiring. Slowly pour in the chicken broth a bit at a time, stirring constantly to keep the mixture smooth. Let it simmer gently until slightly thickened.
Return the chopped chicken to the pot and bring the mixture to a gentle simmer. Let it simmer uncovered for 25 minutes, so the flavors deepen and everything gets yummy.
While the soup simmers, make the dumpling dough. In a bowl, whisk together the flour, baking powder, and salt. Stir in the half-and-half and fresh parsley (if using) until just combined. Don’t overmix. It should be thick and sticky.
Drop spoonfuls of dumpling dough on top of the gently bubbling soup, spacing them slightly apart. Cover with the lid and cook for 15 minutes. I know it is hard, but no peeking! The steam will cook the dumplings until they’re light and fluffy.
After 15 minutes, uncover and check a dumpling if they are cooked. Check if the center’s cooked through, you’re good to go. If not, cook uncovered for another 2 to 3 minutes.
Ladle into bowls with a dumpling or two in each, plus plenty of chicken and broth. Sprinkle it with extra parsley if you like, and enjoy!