To make bibimbap, you don't need a lot of time -- but you do need lots of small bowls! It is beautifully served as a composition of the separate ingredients, which are balanced to lend an assortment of flavors and textures, from grassy to sweet and chewy to crisp. The runny yolk on top, when pierced, serves as a rich sauce that unites the entire dish.
Place the sesame seeds in a dry skillet over medium-high heat and toast until golden, 3 to 4 minutes. Transfer to a small bowl.
Cook the brown rice according to the package directions, omitting any fat. Once cooked, leave the rice covered in the pot (off the heat).
Meanwhile, combine the soy sauce, ginger, sugar, garlic and 1 teaspoon sesame oil. In separate bowls, combine the meat and scallions with 1/4 cup of the soy mixture; the cucumbers with 2 teaspoons; the carrots with 2 teaspoons; the zucchini with 2 teaspoons and the shiitakes with the remaining 1 teaspoon of the soy mixture.
Heat the remaining tablespoon sesame oil in a large nonstick pan over medium-high heat. Add the meat and stir-fry until it is slightly pink in the center, 2 to 3 minutes. (It will continue cooking as it stands.) Use tongs to transfer the meat to a clean bowl, leaving the liquid in the pan.
Add the spinach to the pan and cook, tossing constantly, until it is soft and bright green, about 2 minutes; transfer to a clean bowl. Add the zucchini and stir-fry until crisp-tender, 1 to 2 minutes; transfer to a clean bowl. Add the shiitake and stir-fry until the liquid is thick and the mushrooms are tender, about 3 minutes. Add the eggs to the pan and cook until the whites are just set.
Divide the rice evenly among 4 serving bowls; top with the beef, cucumbers, carrots, zucchini, spinach and shiitake. Drizzle with gochujang and sprinkle with the sesame seeds. Add an egg to the center of each bowl and drizzle with more gochujang. Serve immediately.