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Cut each rice cake crosswise into about 3-inch long pieces. Then cut each piece into quarters. (Day old rice cake is best for cutting.) Boil the rice cake in a small pot of boiling water until softened. Drain, and mix with the soy sauce, sugar and sesame oil, and let sit for about 30 minutes.
Cut the beef into about 3.5-inch long strips to create 8 pieces. (Cut the beef longer than the rice cake because the beef will shrink more.) Combine with the soy sauce, sugar, sesame oil, rice wine, minced garlic, and pepper. Marinate for about 30 minutes.
Cut the mushrooms lengthwise into about 8-centimeter long strips. Blanch them in salted boiling water just to soften, about a minute. Drain.
Cut the green parts of scallions into 8 3-inch long pieces. Cut the carrots into 2 3-inch long pieces. Then cut each one into quarters.
Heat a large pan with a little bit of oil over medium low heat. Place the skewers in the pan. Cook each side until the meat and vegetables are cooked through and slightly caramelized, 1 – 2 minutes each side. Garnish with the optional sesame seeds or pine nuts.