Soak, drain and rinse the rice in a strainer until the water runs clear and is not cloudy. Combine the rinsed rice with 3/4 cup water in a small saucepan. Bring to a simmer over medium heat, then over and turn the heat down to low. Cook, undisturbed, until the water is absorbed, about 15 minutes. Remove from the heat and let sit, covered, for another 15 minutes. Uncover and fluff the rice with a fork. Stir in 1 tablespoon sesame oil, the sesame seeds, 1 tablespoon soy sauce, the vinegar, sugar and 1/2 teaspoon salt. Let cool completely (the rice can be refrigerated up to 1 day ahead).
Combine the baby spinach with 1 tablespoon water in a microwave-safe bowl. Cover and microwave until wilted, 3 to 4 minutes. Drain and toss with the remaining sesame oil and soy sauce. Set aside.
Preheat a waffle iron to medium-high. Generously brush the top and bottom of the waffle iron with sesame oil. Stir the 2 beaten eggs into the cooled rice. Evenly spread about 1/2 cup of the rice mixture in the waffle iron. Close and cook until the rice is golden in spots and can be easily lifted out with a spatula, 4 to 6 minutes. Repeat with the remaining rice mixture. Transfer to two plates and set aside.
Lightly brush the waffle iron bottom with more sesame oil. Crack the remaining 2 eggs right onto the waffle iron and keep the iron open. Sprinkle the eggs with salt. Cook until the whites are set and the yolk is still runny, 4 to 5 minutes. Serve an egg on top of each rice waffle.
Top with the spinach, kimchi, gochujang paste and sesame seeds.
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