Put water and kombu in a pot and let it sit for at least 30 minutes until the kombu softens.
Heat the pot over very low heat for about 10 minutes or longer. When small bubbles start to appear on the surface, remove the kombu.
Increase the heat and bring the water (kombu dashi) to a boil. Once boiling, reduce the heat to low, add bonito flakes, and let it simmer for 3 minutes.
Turn off the heat and strain the mixture through a sieve lined with paper towels or a cloth (such as cheesecloth). Alternatively, if you don't mind a few fine bonito flakes remaining, you can simply use a fine-mesh strainer. Awase dashi, which combines kombu dashi and bonito dashi, is now ready.
Cut the cabbage into bite-sized pieces. Thinly slice the green onions.
Return the dashi to the pot and bring it back to a boil. Once boiling, gently crack the eggs into the pot. Let them simmer over low heat for 3 minutes.
Add the cabbage to the pot and continue cooking for another 30 seconds.
Turn off the heat and dissolve miso in the pot using a ladle and chopsticks, being careful not to break the eggs. Serve in individual bowls and top with the green onions.