Heat a large (12-inch) heavy cast-iron skillet over medium heat. Drizzle the sesame oil into the pan and gently spread the rice in a loose layer.
For the bibimbap sauce: Mix together all the sauce ingredients in a small bowl until smooth and set aside.
For the toppings: Mix together the first 6 ingredients in a small bowl to make the seasoning sauce. Set aside by the stove.
Drizzle 1/2 teaspoon oil in a medium (9-inch) nonstick skillet over medium-high heat, add the bean sprouts and 1 teaspoon seasoning sauce and saute until crisp-tender, about 30 seconds. Arrange nicely on the rice. Cook the remaining toppings in the same manner using the following measurements and times and arranging each on top of the rice: carrots, 1/2 teaspoon oil, 1 teaspoon sauce, until crisp-tender, 30 seconds; zucchini, 1/2 teaspoon oil, 2 teaspoons sauce, until just tender, about 1 minute; snow peas, 1/2 teaspoon oil, 2 teaspoons sauce, until crisp-tender, 45 seconds; spinach, 1 teaspoon oil, 1 tablespoon sauce, until just wilted, 30 seconds; mushrooms, 4 teaspoons oil, 1 tablespoon sauce, until tender, 1 minute; beef, 2 teaspoons oil, stir in the sugar and remaining sauce, until cooked through, 1 to 2 minutes. Spoon the juices over the meat on the rice. Deglaze the pan with mirin as necessary.
Arrange the kimchi on the rice. For a more attractive arrangement, place the ingredients so that similar colors are separated. Spoon the bibimbap sauce on the center, or serve on the side, if you like. Make a little well in each of the mushrooms, meat and spinach. Gently place an egg yolk into each well. Garnish with black sesame seeds and micro cress. Bring the whole pan to the table and mix tableside.
Cook in a pot, like paella. The sauteing in this dish is very quick so be sure you have all the toppings ready before you start cooking.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.