Marinate the beef. Toss the sliced flank with the soy sauce, brown sugar, 1 tbsp sesame oil, 3 cloves of the minced garlic, the ginger, and a few grinds of black pepper. Let it sit while you prep everything else, at least 15 minutes. Slicing is easier if you freeze the steak for 15 minutes first, and cutting across the grain keeps the cooked beef tender.
Make the gochujang sauce. Stir the gochujang with the sugar, 1 teaspoon of the sesame oil, and 2 to 3 tablespoons of water until it is smooth and pourable. Loosening the paste this way lets it coat the whole bowl instead of sitting in one hot clump.
Blanch the spinach in boiling water for 30 seconds, then lift it out into cold water. Squeeze it dry hard with your hands, chop roughly, and season with a pinch of salt, half a clove of minced garlic, and a few drops of sesame oil. Squeezing matters, since wet spinach waters down the bowl.
Blanch the bean sprouts in the same water for 2 minutes, drain well, and season the same way with salt, garlic, and a little sesame oil.
Heat a little of the remaining sesame oil in a skillet over medium heat. Saute the carrot matchsticks for 2 minutes until just softened, season with a pinch of salt, and set aside. Add the mushrooms with the last of the garlic and cook 4 to 5 minutes until browned, then set aside.
Turn the heat to high and add the neutral oil. Sear the beef in a single layer, working in two batches so the pan stays hot. Let each batch brown for about a minute before stirring, 3 to 4 minutes total per batch. Crowd the pan and the beef steams and turns gray instead of searing.
Fry the eggs in a little oil to your liking, keeping the yolks runny so they mix into the rice.
Assemble. Divide the warm rice among four bowls. Arrange the beef and each vegetable in its own section on top, set a fried egg in the middle, and add a spoon of the gochujang sauce. Finish with sesame seeds and spring onion. Mix everything together at the table before eating.