Bring rice and 1 3/4 cups water to a boil in a saucepan. Cover, reduce heat to low and cook for 18 minutes; rest covered off heat for 10 minutes. Mix gochujang, vinegar, honey, warm water and 1 teaspoon sesame oil in a small bowl.
Heat 1 teaspoon neutral oil in a large skillet over medium-high heat. Wilt spinach for 1 to 2 minutes with a pinch of the measured salt and transfer to a plate.
Add another teaspoon oil and cook carrots for 3 minutes with another pinch of salt; transfer to the plate. Add another teaspoon oil and cook bean sprouts for 4 minutes until steaming hot and softened, then transfer to the plate.
Add 1 teaspoon oil and cook mushrooms for 4 to 5 minutes with the remaining salt, then transfer to the plate. Keep the vegetable portions separate.
Heat 2 teaspoons oil in the skillet. Brown beef for 7 to 8 minutes, breaking it into fine crumbles and cooking to 160°F. Stir in garlic, soy sauce and sugar for 1 minute, then remove from heat and add the remaining teaspoon sesame oil.
Heat the remaining tablespoon neutral oil in a separate nonstick skillet. Fry eggs for 3 to 4 minutes until whites are fully set, covering briefly as needed; cook yolks firm if preferred.
Divide rice among four bowls, arrange beef and vegetables over it and put one egg on each. Add the sauce at the table and stir the components together before eating.