Heat a large heavy-bottomed skillet (preferably cast iron) over medium heat. Add 1 tablespoon of the sesame oil and gently spread the rice over the bottom of the skillet in a loose layer. Cook, undisturbed, until the bottom of the rice develops a golden crust, 8 to 10 minutes. This should be about the same time the toppings are done, but if the rice is ready beforehand, just turn off the heat.
Meanwhile, in a small bowl, stir together the soy sauce, 2 tablespoons of the mirin, the remaining 1 tablespoon sesame oil, the garlic, ginger, and sesame seeds. Mix 2 tablespoons of the sauce with the beef and sugar and set aside. Set the remaining seasoning sauce by the stove.
As each vegetable topping is finished, arrange it on a section of the rice in the skillet so the toppings resemble the spokes of a bicycle wheel. In a medium nonstick skillet, heat 1/2 teaspoon of vegetable oil over medium-high heat. Add the bean sprouts and 1 teaspoon of the seasoning sauce and cook, stirring, until crisp-tender, about 30 seconds. Arrange the bean sprouts on a section of the rice. Heat 2 teaspoons of oil in the skillet, add the beef, and cook until cooked through, 1 to 2 minutes. Arrange the beef on the rice.
Heat 1 tablespoon of oil in the skillet, add the mushrooms and 1 tablespoon of the sauce, and cook until tender, 1 minute. Arrange the mushrooms on the rice. Heat 1/2 teaspoon of oil in the skillet, add the zucchini and 2 teaspoons of the sauce, and cook until just tender, about 1 minute. Arrange the zucchini on the rice.
Heat 1/2 teaspoon of oil in the skillet, add the snow peas and 2 teaspoons of the sauce, and cook until crisp-tender, 45 seconds. Arrange the snow peas on the rice.
Heat 1/2 teaspoon of oil in the skillet, add the carrot and 1 teaspoon of the sauce, and cook until crisp-tender, 30 seconds. Arrange the carrot on the rice. Finally, heat 1 teaspoon oil in the skillet, add the spinach and 1 tablespoon of the sauce, and cook until just wilted, 30 seconds. Arrange the spinach on the rice.
Deglaze the pan with the remaining 1 tablespoon mirin, scrape up any browned bits from the bottom, and spoon the juices from the skillet over the meat on the rice. Arrange the kimchi on a section of the rice.
Spoon the Gochujang Sauce on the center of the rice or serve on the side, if you like. Make a little well in the spinach, mushrooms, and beef and gently put an egg yolk into each or arrange the fried eggs, if using, over the dish. Sprinkle the Bibimbap with black sesame seeds and radish sprouts, if desired. Bring the skillet to the table, set it on a trivet, and mix everything together before spooning into bowls.