Make the marinade for the beef. Combine the soy sauce, sugars, garlic, green onions, sesame seeds in a large bowl; add the sliced beef strips to the marinade, and season with salt and pepper. Cover, and refrigerate for at least 2 hours.
Cook rice in your rice cooker.
Combine shiitake mushrooms and 1/2 cup hot water in a small bowl, and soak for about 10 minutes, until pliable. Trim off and discard the stems. Thinly slice the caps. Set aside.
Bring a saucepan of water to a boil. Add spinach to the water just long enough to wilt the leaves, and then drain and pat dry. Set aside. Combine cucumber and carrots in a bowl, and season with salt and pepper. Set aside.
Preheat wok over medium-high heat. Cook carrots and cucumbers in a small amount of sesame oil to soften, stirring frequently. Remove from pan, and set aside. Add a small amount of sesame oil to the pan, and cook spinach in sesame oil for a minute or two. Remove spinach from pan,and set aside. Add the meat strips and marinade to the wok; cook, stirring frequently, until the liquid reduces in volume, about 4 to 5 minutes.
Divide the rice into the bowls, and gently pack to the bottom. Arrange the cucumbers and carrots, bean sprouts, greens, shiitake mushrooms, and beef mixture over each portion of rice. Immediately before serving, add one cooked egg (be sure the yolk is soft), drizzle with about a tablespoon of sesame oil, and top with the nori. Serve Kochujang sauce as a condiment.