Cook the rice in a rice cooker or following package directions, using a little less water than called for. The rice for bibimbap should be a little drier than usual for best results.
Chicken: Cut into thin 2-inch long strips. Mix in 1 tablespoon of soy sauce, 2 teaspoons of sugar, 2 teaspoons of sesame oil, 2 teaspoons of rice wine, 1 tablespoon of chopped scallion, 1 teaspoon minced garlic, 1/2 sesame seeds and a pinch of pepper. Marinate for 20 minutes. Sauté in a skillet for 2 - 3 minutes over high heat.
Soy bean sprouts: Bring 1 cup of water to a boil with a teaspoon of salt. Add the bean sprouts and boil for 3 minutes. Drain quickly and shock in cold water to stop cooking. Drain again. Toss with 1 teaspoon of minced garlic, 2 teaspoons of sesame oil, 1/2 teaspoon of sesame seeds, and salt and pepper to taste.
Spinach: Blanch the spinach in salted boiling water only until wilted, 30 - 40 seconds. Drain quickly and shock in cold water. Squeeze out water. Cut into 3-inch lengths. Toss with 1 tablespoon of chopped scallion, 1 teaspoon minced garlic, 1 teaspoon of sesame oil, 1/2 teaspoon of sesame seeds and salt and pepper to taste.
Cucumbers: Cut the cucumbers in half lengthwise and then thinly slice crosswise. Generously sprinkle salt over sliced cucumbers and set aside for 10 - 15 minutes. Squeeze out excess liquid. Toss with 1 tablespoon of chopped scallion, 1/2 teaspoon minced garlic, 1 teaspoon sesame oil and 1/2 teaspoon of sesame seeds.
Zucchinis: Cut the zucchinis in half lengthwise and then thinly slice crosswise. Generously sprinkle salt over sliced zucchinis and set aside for 10 - 15 minutes. Squeeze out excess liquid from salted zucchini by hand. Add 1 tablespoon of chopped scallion, 1/2 teaspoon minced garlic, 1 teaspoon sesame oil and 1/2 teaspoon sesame seeds. Sauté in a lightly oiled skillet for 1 - 2 minutes over medium high heat.
Carrots: Julienne the carrots into match sticks. Sauté in a lightly oiled skillet for 1 - 2 minutes over medium high heat, sprinkling salt and pepper to taste.
Combine all of the sauce ingredients in a small bowl and mix thoroughly.
Assemble: Place a serving of rice in a big bowl. (If using a stone bowl, heat it over medium heat until very hot. Add 1 tablespoon of the sesame oil and rice and cook the rice for several minutes until the rice sizzles.) Nicely arrange a small amount of each prepared vegetable and beef over the rice. Drizzle a little sesame oil over. Top with an egg fried sunny-side up and serve with the red pepper paste sauce.