Adapted from a recipe by Nick Kindelsperger at Serious Eats. If served in a super-hot cast iron or stone casserole, this is called "dol sot bibim bap," and will form a deliciously crunchy layer on the bottom of the dish...
In a bowl, stir together soy sauce, 1 tablespoon sesame oil, sugar, garlic, ginger and apple juice. Add beef, stir well, and set aside.
Cook rice according to package directions.
In a skillet, heat 1/2 tbsp sesame oil. Saute zucchini slices until tender, about 5 minutes. Transfer to a bowl. Add 1/2 tbsp oil to skillet. Saute mushrooms until tender, about 5 minutes. Add to zucchini.
Blanch bean sprouts in boiling water for 1 minute; drain well.
Cook beef in same skillet (do not add more oil) until fully browned. Set aside.
Heat remaining tablespoon sesame oil in skillet. Add eggs and fry until whites are set but yolks remain runny.
In a large bowl, place half the rice in a mound in the center. Add half the zucchini, half the mushrooms, half the bean sprouts, and half the carrots around the outside. Sprinkle half the beef over the vegetables. Top with a fried egg. Serve with chili paste and vinegar to taste. When ready to eat, break the yolk and stir everything together thoroughly.