Soak 5 g dried kelp (kombu) and 10 g dried shiitake mushroom in 500 ml water for at least 30 minutes, preferably overnight.
Bring a small pot of water to a rolling boil. Drain 150 g konjac (konnyaku) and boil it for 2-5 minutes, then drain and leave to cool. Prepare the vegetables and tofu according to the ingredients list. When the konjac is cool enough to touch, tear it by hand or use a spoon to scoop rough chunks.
Heat a large deep pot over medium and add 1 tsp toasted sesame oil. Once fragrant, add the konjac and stir fry for 1 minute. Add the vegetables one at a time, stir frying for 1-2 minutes between each addition. Start with 100 g burdock root (gobo), then 75 g carrot, then 250 g daikon radish.
Remove the rehydrated shiitake from prepared dashi and cut into thin slices. Add it to the pot along with 20 g fried tofu pouch (aburaage), 50 g shimeji mushrooms, and 100 g enoki mushrooms. Pour the dashi into the pot and bring to a gentle simmer, removing the kombu just as small bubbles start to appear. Simmer for 10-15 minutes, or until the carrot and daikon are fork tender.
Stir in 2 tbsp sake, then mix and add 2 tbsp mirin. Increase the heat and let it boil for 1-2 minutes, then reduce the heat back down and add 3 tbsp Japanese soy sauce (koikuchi shoyu).
Add 50 g Japanese leek (naganegi) to the pot and simmer on low for 1-2 minutes. Turn off the heat, taste test and add a splash of soy sauce or water to balance the flavor to your preference.
Divide the soup between serving bowls, then top with a drizzle of toasted sesame oil and a sprinkle of finely chopped green onions. Enjoy!
Notes
Cold-soak the dashi, never hot. Leave the kombu and dried shiitake in cold water in the fridge overnight (or at least 30 minutes at room temp).