Add potatoes to a large pot of salted water, and bring to a boil; cook until tender, about 15 minutes. Drain, pour potatoes into a large bowl. Stir in butter, and mash with a fork or potato masher. Set aside.
Heat 1 tablespoon oil in a large skillet over medium heat. Stir in onions, and cook until soft. Stir in beef and soy sauce. Cook, stirring continuously, until beef is browned and all the liquid is evaporated. Combine beef and onions with potatoes, and stir well.
Heat 1/2 cup oil in a deep skillet or wok over medium-high heat.
Shape potato and beef mixture into 10 similar-size balls, and flatten into patties. Dredge patties in flour. Then dip in egg, and coat with panko bread crumbs. Carefully place patties into oil, and fry until golden brown on each side.
