Pat short ribs dry and season all over with salt and pepper. Heat a Dutch oven (or a large oven-safe high-walled skillet) over medium-high heat, then add butter and olive oil. Sear short ribs 2-3 minutes on all sides until a deep golden brown. Transfer to a plate and set aside.
In remaining drippings over medium heat, saute carrot and onion until vegetables begin to soften, about 3 minutes. Add garlic and saute 1 additional minute until fragrant. Pour apple cider and beef broth into pan and bring to a simmer.
Nestle short ribs back into Dutch oven and arrange cranberries, apple slices, thyme, and rosemary around shortribs. Cover Dutch oven and place in 300F oven and let cook 2 hours or until short ribs are very tender.
Remove short ribs and carrots from Dutch oven and transfer to a plate. Remove and discard thyme/rosemary sprigs.
Whisk flour with a little bit of braising liquid from Dutch oven until smooth. Add flour mixture back to pan, stirring until smooth. Bring to simmer over medium-high heat until thickened. Return short ribs and carrots back to Dutch oven. Serve short ribs warm with mashed potatoes. Enjoy!