First, steam the rice. Bring water to a boil in a small saucepan, then stir in rice, cover, and turn heat to low. Steam for about 20 minutes, or until the water has been absorbed and the rice is fluffy.
Cook the veggies. Heat about 1 tbsp of the vegetable oil in a large cast iron pan over medium-high heat. Once it's hot, add onion, chili peppers, bell pepper, and carrots and sauté until slightly softened, about 2 to 3 minutes. Next, add the cucumber and mung bean sprouts. Sauté until soft, about 3 to 4 minutes. Then add spinach, sesame oil, and soy sauce and cook for another minute or two until the spinach is wilted. Finally, transfer the contents of the pan to a bowl and set aside.
Next, crisp the rice. Add about 1 to 2 tbsp more vegetable oil to the cast iron pan and return to medium heat. Carefully swirl the oil around in the pan to evenly coat the bottom and sides. Then, add the steamed rice, pressing it into the bottom and sides of the pan to create a "bowl" shape for the veggies to sit in.
Add the cooked veggies back into the pan, on top of the rice. Cook for about 5 minutes or so, to allow the rice enough time to crisp.
In the meantime, fry eggs to your liking in a separate pan.
Finally, add bibimbap sauce to the veggies and carefully stir to incorporate, without disturbing the rice. Top with fried eggs, then serve family-style.