01 - Heat unsalted butter and olive oil in a large deep skillet over medium-high heat. Add chicken pieces and brown on both sides until golden, about 4 minutes per side. Remove chicken and set aside.
02 - In the same skillet, add diced onion, celery, and carrots. Cook, stirring occasionally, until softened, about 5 minutes. Add minced garlic, salt, black pepper, dried thyme, and smoked paprika; cook for 1 minute, stirring until fragrant.
03 - Sprinkle all-purpose flour over the vegetables and stir to coat evenly. Gradually pour in the chicken broth while stirring constantly to form a smooth mixture without lumps.
04 - Return browned chicken pieces to the skillet. Bring the mixture to a simmer, cover with a lid, and cook for 10 minutes.
05 - Stir in heavy cream and chopped fresh parsley. Taste and adjust seasoning if necessary.
06 - While the chicken is simmering, combine all-purpose flour, baking powder, baking soda, and salt in a mixing bowl. Cut in cold cubed butter using a pastry blender or forks until the mixture resembles coarse crumbs. Stir in whole milk just until combined, being careful not to overmix.
07 - Drop heaping tablespoons of dumpling dough evenly over the simmering chicken and gravy. Cover the skillet and cook on low heat for 15 to 18 minutes, until dumplings are puffed and cooked through.
08 - Uncover and simmer for an additional 2 to 3 minutes to allow the sauce to thicken slightly. Sprinkle additional fresh parsley over the dish before serving.