Preheat oven to 350 degrees.
Pat short ribs dry with paper towels. Season with salt and pepper. Heat 1 tablespoon oil in a Dutch oven (or other oven-safe pot that has a lid) over medium-high heat. Add half of short ribs and cook until browned on all sides, 2 - 3 minutes per side. Remove and brown second half. Then remove second half of ribs from pot and pour out excess fat.
Heat second tablespoon oil in the same Dutch oven. Add onion and carrot and season with salt and pepper. Cook, stirring occasionally, for 8 minutes. Add garlic and cook 2 more minutes, until garlic is fragrant. Add molasses and stir to combine. Pour in apple cider and scrape up any brown bits stuck to the bottom of the pot. Put short ribs back in the pot over vegetables. Pour beef broth over ribs. Add thyme sprigs. Cover and bring to a simmer over high heat.
Turn off heat and transfer the pot (with lid on) to the oven. Bake until meat is tender, about 2 ½ hours.
Remove short ribs from oven and let rest, covered, for 10 minutes. Skim excess fat off the top. (If you are making ahead, refrigerate and then skim the solidified fat off the top before reheating.)
While ribs are resting, sauté the apples. Melt butter in a pan over medium heat. Add apples and cook, stirring occasionally, for 10 minutes. Serve ribs topped with apples.