Season turkey with salt and pepper. Heat oil in a large heavy pot over medium heat. Working in batches, cook, turning occasionally, until browned on all sides, 15–20 minutes per batch; transfer to a large plate.
Add onion, leeks, celery, and garlic to pot and cook, stirring occasionally, until softened, 5–8 minutes. Add wine, parsley, and thyme and sage sprigs; bring to a boil, reduce heat, and simmer until reduced by half, 8–10 minutes.
Return turkey to pot and add broth. Bring to a boil. Reduce heat, cover pot, and simmer until meat is tender and cooked through and liquid is reduced by half, 2 1/2–3 hours. Add carrots and cook, uncovered, until carrots are soft and meat is falling off the bone, 35–45 minutes; add butter season with salt and pepper.
Transfer turkey and carrots to a platter. Strain sauce; serve alongside. Top with chopped parsley.