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Put the dried shiitake in a heatproof bowl and pour over 300 ml plus 1 teaspoon of just-boiled water from a kettle and leave to hydrate for 30 minutes. Strain the softened shiitake, reserving the soaking liquid. Put the shiitake, shallots, garlic, ginger and 1 tablespoon water in a blender or food processor and blitz to a paste.
Heat the sesame oil in a small frying pan over a low heat and fry the shiitake paste for 3 minutes, stirring often. Tip the mixture, including any oil, into the slow cooker pot with 900ml of just-boiled water from the kettle, the soy sauce and white part of the spring onions and stir until combined. Cover and cook for 2-2½ hours on low or 1½-2 hours on high.
15 minutes before the end of the cooking time, mix together the miso and tahini and stir into the pot with the sugar snap peas and kale. Cover and finish cooking until the kale has wilted.
Divide the cooked noodles between four large serving bowls. Ladle over the vegetables and broth and top with the carrot, chilli, sesame seeds, green part of the spring onions, dulse or nori and an extra drizzle of sesame oil.