Begin by making the pork: Preheat the oven to 375 degrees. Mix together the rest of the ingredients and brush them all over the pork. Place the pork on a foil lined baking sheet. Bake the pork for thirty minutes, or until it reaches an internal temperature of 145 degrees. Let it sit for ten minutes and then slice it in thin slices.
While the pork cooks, prepare the rest of the ingredients. Make the gochujang sauce: In a jar combine the gochujang, lemon juice, garlic, two tablespoons of the sugar and a quarter cup of the vinegar.
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 16 minutes. Then, remove it from the heat and let it sit covered for ten minutes.
Next, pickle the radishes and cucumbers. Sprinkle each with a quarter teaspoon salt and let it sit ten minutes. Squeeze each vegetable 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 soy sauce and sugar, and a teaspoon of olive oil until they're crisp tender. Remove and keep warm.
Add the brussels sprouts to the pan along with a tablespoon of butter and cook about five minutes on medium heat, until the sprouts are tender. Add one teaspoon of sesame oil and one teaspoon of soy sauce.
Fry four eggs in a tablespoon of butter. Put the rice in bowls and top each portion with some pork slices, an egg, some carrots and brussels spouts, some pickled carrots and radishes. Drizzle the gochujang sauce over everything.