In a blender, blend the soy sauce, mirin, sake, black pepper, daikon radish, Asian pear, apple and garlic until smooth.
Rub the short ribs with the vegetable oil and season with ample kosher salt and freshly ground black pepper. Set a large cast-iron pan over high heat, and, when very hot, sear the ribs until golden-brown on all sides, 3 to 5 minutes per side. Only sear as many ribs as will fit comfortably in the pan; work in batches if necessary. Resist the urge to turn the meat before each surface has formed a crust. Trust us, it will pay off in the end! Drain the seared ribs of their fat.
Place the seared meat at the bottom of a slow cooker. Add the potatoes, onion, carrot and mushrooms on top. Cover with a layer of rice cakes then pour in soy sauce mixture and stock on top. The sauce may have settled, so give it a good stir before pouring.
Cook on the slow cooker's low setting until the meat is fork tender and falls off the bone, about 6 hours.
Alternatively, if not using a slow cooker: If you're not down with the electric cooker, that is very much OK. Sear the meat in a Dutch oven, drain the fat, and layer in the vegetables, rice cakes, sauce and beef stock and bring to a boil. Reduce the heat to very low and simmer gently, stirring occasionally, until the meat is fork tender, about 2 hours.