Soak cut gobo in a bowl of water. You will notice the water get very brown after a few minutes. Soak until ready to cook.
Boil konnyaku for 2 minutes or pour boiling water over konnyaku and let it soak for a few minutes. Drain and rinse. Thinly slice.
Boil abura-age for 1 minute or pour boiling water over it and let it soak for a minute. Drain and quickly rinse under cold water. When cool enough to handle, gently squeeze out excess water. Thinly slice.
Heat sesame oil in a pot over medium heat. Drain and rinse gobo, then add to the pot along with daikon and carrots. Stir to coat with oil and cook for a minute.
Stir in cabbage and cook another minute.
Stir in konnyaku and abura-age. Add dashi and mirin and bring to a boil. Lower heat to a simmer and cook for about 10 minutes or until all the veggies are tender.
Turn off heat and add green onions. Place miso in a ladle and fill with some of the broth. Stir until miso dissolves in the broth, then mix into the soup.
Notes
1) You can swap for 1/2 a block of Japanese soft tofu or silken firm tofu*. Cut into cubes or just break into chunks with your hands. 2) You can use 1 teaspoon instant dashi granules* + 700ml water, or make your own kombu dashi. To make kombu dashi, soak about 8 grams of kombu in 700ml of water overnight (covered) in the fridge, or for 2-3 hours at room temperature (mizudashi method). OR you can use the nidashi method and place kombu and water in a pot. Let the kombu soak for 30-60 minutes, then place over medium low heat and slowly bring to a boil. Right before boiling, remove the kombu. You can also soak 2 dried shiitake mushrooms with the kombu, then slice the shiitake and use it in the soup (add with the cabbage).Variations: Feel free to play around with the ingredients! Other common veggies are taro, mushrooms, lotus root, and bamboo. You could also add in a little grated ginger for spice. Get creative!