Season the chicken thigh pieces with ½ teaspoon kosher salt and ¼ teaspoon black pepper.
Then melt 1 tablespoon butter in large Dutch oven over medium high heat. Add seasoned chicken. Cook for 3-4 minutes until it starts to brown, then flip on the other side and continue to cook for another 3 minutes. Brown bits should appear at the bottom of the pot.
Remove the chicken onto a plate and set aside. Add in remaining 3 tablespoons butter, diced onion, celery, and carrot along with 1 teaspoon kosher salt and another ¼ teaspoon black pepper. Cook for about 3-4 minutes until vegetables become translucent around the edges. Do not overcook the vegetables. (Reduce heat if the brown bits start to get too dark to avoid burning.)
Add in minced garlic, thyme, and turmeric and cook for an additional 30 seconds. Slowly pour in chicken stock while scraping the brown bits from the bottom of the pan, incorporating them into the soup. Bring to a boil, then add chicken back to the pot and 1 teaspoon kosher salt. Reduce heat to medium low and continue cooking until chicken is cooked and broth is flavorful, about 8-10 minutes.
Meanwhile, start preparing your dumplings by whisking together the flour, baking powder, salt, black pepper, parsley, and Parmesan into a medium bowl. Stir in the buttermilk, egg, and melted butter until a wet dough forms.
Scoop 2 tablespoon-sized spoonfuls (#40 scoop) of the dumpling dough into the simmering broth. Cover and cook for 15 -18 minutes until the dumplings are puffed and cooked through. (To test the doneness of dumplings, remove one and cut into it to see that it’s fully cooked inside. If not, continue cooking for a couple more minutes.)
Gently stir half and half into soup and let cook another 5 minutes, not allowing the soup to come to a boil. Taste and add additional salt and pepper, if needed, then serve immediately and enjoy!