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1
Combine the kombu and cold water in a pot. Bring slowly over low heat. Remove the kombu just before the water reaches a full boil — the dashi is ready when a faint marine fragrance rises and the water has taken on a pale amber colour.
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2
Heat the sesame oil in a heavy pot over medium heat. Add the gobo first; sauté alone for two minutes before adding the daikon, carrot, satsumaimo, tofu, and konnyaku in succession. Turn frequently. This is where the dish lives or dies: the vegetables must develop a light aromatic crust in the sesame oil — fragrance should be pronounced and nutty — before any liquid is added. Without this step the soup will be flat and thin, no matter how fine the dashi.
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3
Pour the kombu dashi over the sautéed vegetables. Bring to a gentle simmer. Add the mirin and sake. Simmer until the root vegetables are fully tender — a bamboo skewer should meet no resistance at the centre of the daikon and satsumaimo.
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4
Season with light soy sauce, added gradually. Taste: the soup's flavour should rest on clean vegetable sweetness and kombu umami; the soy only lifts and frames. Serve immediately, or reserve — the soup deepens and unifies further on reheating.