Sprinkle the short ribs with salt and pepper. Heat a heavy cast-iron pot or Dutch oven over medium heat until it just begins to give off wisps of smoke; drizzle in about 1 1/2 tablespoon of olive oil. Sear 3 ribs in the hot oil until golden brown on all sides. Heat pot until smoking hot again, drizzle in 1 1/2 tablespoon of olive oil, and sear the remaining ribs as before. Transfer ribs to a plate and set aside. Discard used oil.
Place remaining 1 tablespoon of olive oil in the pot over medium heat, and stir in the onion, celery, carrots, and fennel. Sprinkle with a pinch of salt, and cook until the vegetables are soft and the onions are translucent, stirring frequently, about 8 minutes. Pour in the Burgundy wine; scrape and dissolve all the browned flavor bits from the bottom of the pot. Bring the mixture to a boil, and cook until the liquid is reduced by half, about 10 minutes. Stir in rosemary, thyme, bay leaves, and demi-glace; stir well. Mix in the chicken stock, cherry cola, and balsamic vinegar. Push the short ribs into the vegetable mixture so they are about halfway covered; spread cherries over the ribs. Pour any meat juices over the cherries, bring the short ribs to a boil over medium heat, and cover.
Place the covered pot into the preheated oven, and cook until the meat is tender and falling off the bones, 3 1/2 to 4 hours.
Remove the ribs to a serving platter. Skim fat from the cherry sauce, and strain all the cherries, vegetables, and juices through a fine-mesh sieve into a bowl. Discard the solids, and return the liquid back to the pot. Bring to a boil, and cook the sauce over medium heat until thickened, about 10 minutes, stirring frequently; whisk in butter. Serve sauce over the short ribs.