Prepare and brown the chicken: Pat chicken pieces completely dry with paper towels. Season generously with salt and pepper on both sides. Dredge each piece in flour, shaking off excess. Heat butter and olive oil in a large Dutch oven over medium-high heat. When butter stops foaming, add chicken pieces in batches (don't overcrowd). Brown for 4-5 minutes per side until deeply golden. Remove to a plate and set aside.
Build the aromatic base: In the same pot (don't wipe it out), add diced onion, carrots, and celery. Sauté over medium heat for 5-6 minutes, stirring occasionally, until vegetables soften and onion becomes translucent. Sprinkle in ground thyme and turmeric, stirring for 30 seconds until fragrant.
Create the broth: Pour in chicken broth and apple cider. Use a wooden spoon to scrape up all browned bits from the bottom of the pot (deglazing). Nestle browned chicken pieces back into the pot. Bring to a boil, then reduce heat to maintain a gentle simmer. Cover and simmer for 20-25 minutes until chicken is cooked through and tender (internal temperature of 165°F).
Make the dumpling dough: While chicken simmers, whisk together flour, cornmeal, baking powder, and salt in a large bowl until well combined. Pour in half-and-half and stir gently with a wooden spoon just until mixture comes together into a shaggy dough. Don't overmix — a few lumps are fine. Fold in parsley if using. Let dough rest while finishing the chicken.
Shred chicken and enrich broth: Remove chicken pieces from pot using tongs. Let cool slightly, then shred meat into bite-sized pieces, discarding skin and bones. Return shredded chicken to pot and stir in heavy cream. Taste and adjust seasoning with salt and pepper as needed.
Drop and cook the dumplings: Bring mixture back to a gentle simmer. Using a large spoon or small ice cream scoop, drop heaping tablespoon-sized portions of dumpling dough onto the surface of simmering liquid, spacing them evenly (they'll expand). You should get 12-15 dumplings. Cover pot with lid slightly askew (leave small gap for steam) and simmer gently for 15 minutes without lifting the lid. Dumplings should puff up and look fluffy. Test doneness by inserting a toothpick into center — it should come out clean.
Rest and serve: Turn off heat and let pot sit covered for 10 minutes. This allows dumplings to finish cooking and broth to thicken. Ladle into deep bowls, ensuring each serving gets plenty of chicken, vegetables, and dumplings. Serve immediately.