In a large, high-sided pan, heat the butter and olive oil over medium heat. Season both sides of the chicken pieces evenly with the salt and pepper, then add to the pan, skin side down. Cook chicken until skin is golden, about 5-7 minutes. Flip chicken pieces, and cook for an additional 2 to 3 minutes on the other side. Remove chicken pieces from the pan, and set aside. Add the onion and garlic to the pan and cook, stirring often, until the onion is soft and fragrant, about 3 to 4 minutes. Add the flour and cook for an additional 1 minute. Pour in the apple cider vinegar and stir quickly, scraping up any browned bits on the bottom of the pan. After about 30 seconds, add the chicken stock and apple cider. Bring to a boil, then reduce heat to low and simmer. Add the chicken pieces back to the pan, as well as the rosemary sprigs (it’s a good idea to tie them together with kitchen twine so they’re easier to fish out at the end). Cover pan, and simmer for 30 minutes. Remove cover from pan, flip chicken pieces, and continue to simmer, uncovered, until chicken is very tender, about 30 minutes more. Serve immediately atop mashed potatoes.