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.
Bring the rice and water to a boil in a saucepan over high heat. Reduce heat to medium-low and cover; simmer until the rice is tender and the liquid has been absorbed, 20 to 25 minutes.
Preheat an oven to 425 degrees F (220 degrees C), and place 6 Korean stone bowls in oven. 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.
Transfer the stone bowls from the oven to suitable heat resistant surface. Brush each bowl with sesame oil to coat. Divide the rice into the bowls, and gently pack to the bottom (the rice should sizzle as you arrange). Arrange the cucumbers and carrots, bean sprouts, greens, shiitake mushrooms, and beef mixture over each potion of rice. Immediately before serving , add one raw egg yolk to each bowl, drizzle with about a tablespoon of sesame oil, and top with the nori. Serve Kochujang sauce as a condiment.