Thinly slice 2 zucchinis and place in a small bowl. Season with a fat pinch of kosher salt, then massage zucchinis with your hands, gently at first, then more aggressively as you go, just to soften. Pour off any excess liquid, then add a splash of unseasoned rice vinegar. Taste and season with more salt and vinegar if needed. (You can also add a pinch of sugar or a bit of honey to taste.) Set aside.
Trim roots from 4 scallions; discard. Thinly slice scallions crosswise. Set aside a handful of the dark green tops for serving.
Heat 2 Tbsp. vegetable oil and 2 tsp. toasted sesame oil in a large nonstick skillet over medium-high. Add remaining scallions and cook, stirring often with a heatproof rubber spatula, until starting to soften, about 3 minutes.
Add 4 cups cooked white rice, chilled overnight, and 1½ tsp. kosher salt to scallions and mix well with spatula, making sure there are no lumps of rice remaining. Press firmly into an even, compact layer. Cook until deeply browned underneath, 6–8 minutes.
While the rice mixture is cooking, make the dressing. Whisk together 2 Tbsp. gochujang, 2 Tbp. soy sauce, 2 tsp. honey, remaining 4 tsp. toasted sesame oil, and remaining 2 tsp. unseasoned rice vinegar in a small bowl. Taste for heat: You can add more gochujang if it’s not spicy enough or more water if it’s too fiery. Set dressing aside.
Heat 1½ tsp. vegetable oil in same skillet and heat over medium-high. Crack 2 large eggs into each side of skillet and cook until eggs are setting underneath but whites are still clear, about 1 minute. Sprinkle ¼ tsp. soy sauce over each egg, then continue to cook, basting whites with oil (tilt the pan so that it pools), until completely set and opaque, about 1 minute more.