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Combine kombu and 4 cups water in a large pot. Let sit until kombu softens, 25–30 minutes. Bring to a simmer over medium heat. Immediately remove from heat once water starts simmering; fish out kombu and discard. Add bonito flakes and stir once to submerge them. Return to a gentle boil, reduce heat, and simmer about 5 minutes. Remove from heat and let steep 15 minutes (this ensures you get the most flavorful broth, or dashi, possible).
Meanwhile, cook eggs in a medium pot of boiling water 6 minutes. Transfer to a bowl of ice water (ice bath) and let sit until cold, about 2 minutes. Peel eggs; set aside.
Strain dashi through a fine-mesh sieve into a medium bowl. Discard solids, wipe out pot, and return dashi to pot. Add kimchi and tofu and bring to a very gentle simmer. Remove from heat. Submerge sieve in liquid, add miso and gochujang to sieve, and stir to liquefy both, then press through strainer until pastes are dissolved.
Divide soup among bowls. Top with sesame seeds and drizzle with sesame oil. Cut eggs in half and add to bowls.