Begin by marinating the chicken: In a large bowl mix together the gochujang, soy sauce, sesame oil, brown sugar, garlic, and sesame seeds. Reserve two tablespoons of the marinade in a small bowl. Add the chicken to the big bowl of marinade and coat it all over with the marinade. Cover and refrigerate for two hours.
Preheat the oven to 375 degrees. Spread the chicken out on a foil lined sheet and bake it for 30 minutes. Then, brush the chicken with the reserved marinade and broil for a few minutes.
While the chicken cooks, prepare the rice: In a three- quart saucepan bring a cup and a half of water to a boil. Add the rice and a half teaspoon of kosher salt. Cover and simmer for 16 minutes. Remove from the heat and let it sit covered for ten minutes.
Next, prepare the radishes and cucumbers. Sprinkle each with a quarter teaspoon of kosher salt and let sit ten minutes. Squeeze each dry. Stir in one tablespoon of rice vinegar and one teaspoon of sugar to each vegetable.
Stir fry the carrots in a small skillet with a teaspoon each of soy sauce and sugar, in a teaspoon of olive oil until crisp tender. Remove and keep warm.
Add the spinach to the pan along with the butter and cook until it's wilted. Add one teaspoon of sesame oil and a tablespoon of soy sauce.
Fry four eggs. Serve the eggs, chicken, and vegetables on top of the rice.