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 (for 2 servings). When small bubbles appear on the surface of the liquid, 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 having some 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 cabbage into bite-sized pieces. Thinly slice green onions.
Return the dashi to the pot and bring it 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 an additional 30 seconds.
Turn off the heat and dissolve the miso in the pot using a ladle and chopsticks, being careful not to break the eggs. Serve in each bowl and top with the green onions.