Prepare the sauce by combining all ingredients in a small bowl.
Heat a small pan over medium heat. Add oil and cook each of the vegetable toppings in turn, until they just start to become tender, about 6 minutes for the carrots, 5 minutes for the mushrooms and 3 minutes for the bean sprouts. Use salt and pepper to taste.
Fry the eggs on one side until the whites are firm but the yolks are still runny.
Assemble the bowls by distributing the toppings among the serving bowls with rice. Drizzle with Bibimbap Sauce and sprinkle with sesame seeds.