Start the rice and soak the skewers. Cook the rice according to its package and keep it covered. Put wooden skewers in a baking dish of water so they do not burn under the broiler. Metal skewers need no soaking.
Make the sauce. Stir the gochujang, honey, rice vinegar, sesame oil, garlic and water together until smooth and glossy. Spoon 2 tablespoons into a small separate bowl for glazing the skewers. The rest is for the bowls.
Skewer the edamame. Pat the thawed pods dry and toss with the sesame oil and salt. Thread the pods onto the skewers, pushing through the pod near one end, with about 8 pods per skewer.
Heat the broiler. Set a rack 4 to 6 inches (10 to 15 cm) from the broiler element, line a rimmed baking sheet with foil, and heat the broiler on high.
Blanch the bean sprouts. Bring a medium pot of water to a boil. Drop in the sprouts for 1 minute, then lift them out with a spider or drain them. Rinse under cold water, drain well and toss with a pinch of salt.
Cook the spinach. Drop the spinach into the same boiling water for 30 seconds, until bright green and collapsed. Move it to a bowl of cold water, then squeeze it in your fist until no water drips out. Chop roughly and toss with the soy sauce, sesame oil, garlic and sesame seeds.
Cook the carrots. Heat the neutral oil in a skillet over medium heat. Cook the carrots with a pinch of salt for 1 to 2 minutes, until slightly soft but still bright orange.
Dress the cucumber. Toss the cucumber slices with the rice vinegar and a pinch of salt. Drain off any liquid before assembling.
Broil the skewers. Lay the skewers on the baking sheet and broil for 6 to 8 minutes, turning once, until the pods look tender with a few charred spots. Brush with the reserved glaze and broil 1 minute more, until the glaze looks tacky and deep red. Take them out before it darkens to black.
Fry the eggs. Heat the frying oil in a nonstick skillet over medium-low heat. Crack in the eggs and cook 2 to 3 minutes, until the whites are opaque and fully set with lacy edges. Cook longer for a firmer yolk.
Assemble. Divide the rice among four bowls. Arrange the spinach, sprouts, carrots and cucumber in sections on top, set an egg in the center, and lay two skewers across each bowl. Drizzle with the remaining sauce and sprinkle with scallions and sesame seeds. Squeeze the beans from the pods onto the rice, discard the pods, and stir everything together.