Rehydrate the shiitake mushrooms in hot water until soft, approximately 25 minutes. Drain and then toss with 1/2 tablespoon soy sauce and 1/2 tablespoon toasted sesame oil. Set aside.
Heat a large skillet over high heat with just enough oil to coat the bottom. Toss the beef with 2 teaspoons soy sauce, 1 teaspoon of garlic, a pinch of sugar and some black pepper. Sauté the beef for 1-2 minutes (it will cook very fast), then toss with 1 teaspoon of toasted sesame oil and set aside.
Line up 3 bowls to hold the vegetables separately after they're cooked. Sauté the carrots for 30 seconds with a little more oil (if needed) and 1/2 teaspoon of garlic. Place the carrots in one of the reserved bowls. Sauté the zucchini for 30 seconds with 1/2 teaspoon garlic. Place the zucchini in the second reserved bowl and toss with 1/2 teaspoon sesame oil. Turn the heat down to low. Wilt the spinach with 1 teaspoon garlic, 1 teaspoon soy sauce and 1/2 teaspoon sesame oil. Place in the third bowl.
If using dolsot bowls, cover the insides liberally with oil (if using regular bowls, skip to the next step). Divide the rice evenly between the two bowls and then place small mounds of the beef, mushrooms, carrots, zucchini, spinach and bean sprouts on top of the rice. Top with toasted sesame seeds. Place the bowls on the stove over high heat and allow them to get hot. Once the rice begins to make a crackling noise, allow it to cook for 5 minutes.
If using a regular bowl, divide the rice evenly between the two bowls and then place small mounds of the beef, mushrooms, carrots, zucchini, spinach and bean sprouts on top of the rice. Top with toasted sesame seeds.
Cook the egg sunny side up on a lightly oil skillet and place on top of the beef and vegetables.
Serve with Gochujang.
All of these amounts should be viewed as a guide, not set in stone. You can use less rice, more vegetables, etc. Many bibimbap recipes also include daikon radish. I've omitted it here but you can add it using the same method I've described for the carrots and zucchini.