Whisk the gochujang, rice vinegar, honey, sesame oil, and water until smooth. Add water a teaspoon at a time if you prefer a thinner sauce.
Warm a little oil in a skillet over medium heat. Cook the carrot for 2 to 3 minutes until just tender but still bright; transfer to a plate.
Add spinach with a tablespoon of water and cook until wilted, about 1 minute. In a separate pot, blanch the bean sprouts in actively boiling water for 2 to 3 minutes, until fully hot, then drain well.
In the same skillet, cook beef and garlic until browned, breaking it into small pieces. Stir in soy sauce and sesame oil; cook until the beef reaches 160°F (71°C).
Fry the eggs in the remaining oil to your preferred doneness; cook until whites and yolks are firm if following general food-safety guidance.
Divide warm rice among four bowls. Arrange beef, carrot, spinach, sprouts, and cucumber in separate sections. Top each with an egg, sesame seeds, and sauce. Mix at the table before eating.
