Add 5 g dried kelp (kombu) and 10 g dried shiitake mushroom to a pot with 500 ml water. Cover and allow to soak for at least 30 minutes. (If you have time, leave overnight for a deeper flavour.)
Remove the shiitake mushroom and thinly slice it to add back to the soup later.
Place the pot on the stove and heat until the dashi starts to bubble (don't let it boil). When it's nearly boiling, lower the heat to simmer, remove the kombu, and turn off the heat.
Heat up another large pot on medium and add 1 tsp toasted sesame oil. Add all of the ingredients except for green onion (we'll add that later) and stir fry until the root vegetables are softened slightly.
Pour the dashi into the pot with the other ingredients and simmer until the carrot and daikon are soft enough to pierce with a fork.
Add 2 tbsp sake, 2 tbsp mirin and 3 tbsp Japanese soy sauce (koikuchi shoyu) and simmer for 2-3 minutes.
Finally add 50 g Japanese leek (naganegi) and simmer for another 5 minutes. Taste test, you can simmer for longer to make the flavour more concentrated or add a little more soy sauce to taste.
Dish up and enjoy!
Notes
Store leftovers in an airtight container in the fridge and consume within 2-3 days. Reheat in the microwave or on the stove.