Prepare the rice first so it is ready and warm when you assemble the Easy Korean Bibimbap. If you have leftover cooked white rice, gently reheat it and fluff with a fork so the grains are loose.
In a small bowl, combine the ground beef with soy sauce, sesame oil, brown sugar, and minced garlic. Mix well with a spoon or your hands until the beef is evenly seasoned, then set aside to marinate while you prepare the vegetables.
Place the carrot matchsticks, baby spinach, bean sprouts, cucumber matchsticks, and sliced mushrooms in separate small bowls so they are ready to cook and assemble. This makes it easier to build each bibimbap bowl later.
Heat 1 teaspoon of vegetable oil in a large nonstick skillet over medium heat. Add the carrot matchsticks with a small pinch of salt and cook, stirring often, for about 2 to 3 minutes until they are slightly softened but still bright. Transfer the carrots to a plate and keep them in a small pile.
In the same skillet, add the baby spinach with a small splash of water and a tiny pinch of salt. Cook, tossing gently, for about 1 to 2 minutes until the spinach just wilts. Transfer the spinach to the plate next to the carrots, keeping it in a separate pile.
Add the bean sprouts to the hot skillet and cook for about 2 minutes with another small pinch of salt, stirring occasionally, until they are slightly tender but still crisp. Move the bean sprouts to the plate, again keeping them separate from the other vegetables.
Add the sliced mushrooms to the skillet and cook for 3 to 4 minutes, stirring now and then, until they release their moisture and turn lightly browned. Season with a little salt and black pepper, then transfer the mushrooms to the plate with the other vegetables. Leave the fresh cucumber matchsticks raw for a cool, crisp texture.
Add the marinated ground beef to the same skillet and cook over medium-high heat for about 4 to 5 minutes, breaking it up with a spoon, until the beef is browned and cooked through and most of the liquid has evaporated. Taste and adjust the seasoning with a bit more salt or pepper if needed, then turn off the heat.
In a small bowl, stir together the gochujang, water, rice vinegar, and honey until smooth. The sauce should be thick but pourable; if it feels too thick, add a few more drops of water until it reaches a saucy consistency.
Heat the remaining 1 teaspoon of vegetable oil in a clean nonstick skillet over medium heat. Crack in the eggs and cook them sunny-side up for 3 to 4 minutes, or until the whites are set and the yolks are still runny. If you prefer firmer yolks, cook them a little longer to your liking.
To assemble the Easy Korean Bibimbap, divide the warm cooked white rice evenly among four bowls, gently pressing it down to make a base. Arrange small piles of carrots, spinach, bean sprouts, cucumber, and mushrooms on top of the rice in each bowl, leaving space in the center.
Spoon some of the cooked ground beef into the center of each bowl, then place a fried egg on top of the beef. Drizzle each bowl with a generous spoonful of the gochujang sauce and sprinkle with toasted sesame seeds. Serve immediately, inviting everyone to mix everything together before eating so the rice, vegetables, beef, egg, and sauce combine into one flavorful bite.