Boil rice and 1 3/4 cups water in a saucepan. Cover; cook on low 18 minutes. Rest covered off heat 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. Wilt spinach 1 to 2 minutes with a pinch of measured salt; transfer to a plate.
Add another teaspoon oil; cook carrots 3 minutes with another pinch salt. Transfer to plate. Add another teaspoon oil; cook sprouts 4 minutes until steaming hot and softened; transfer to plate.
Add 1 teaspoon oil; cook mushrooms 4 to 5 minutes with remaining salt. Transfer to plate, keeping vegetables separate.
Heat 2 teaspoons oil. Brown beef 7 to 8 minutes, crumbling finely, until 160°F. Add garlic, soy sauce, and sugar for 1 minute. Remove from heat; add remaining teaspoon sesame oil.
Heat remaining tablespoon neutral oil in a separate nonstick skillet. Fry eggs 3 to 4 minutes until whites fully set, covering briefly if needed; cook yolks firm if preferred.
Divide rice among four bowls; arrange beef and vegetables on top, then one egg each. Add sauce at table; mix before eating.