Add dashima and onion to pot with 2 cups of water and bring to a boil. Reduce heat to medium and continue boiling for 5-7 minutes. Remove dashima.
Add zucchini, mushroom, soondubu, garlic, soy sauce and pepper to pot. Cook for another 3-5 minutes. Stir in sesame seeds and boil for another 1-2 minutes.
Whisk one egg into soup and cook for about 1 minute.
Drizzle sesame oil over soup and finish with chopped green onion prior to turning heat off.
Silken/extra soft tofu: You will find two types of soft tofu at the Asian grocery store - packaged in a square container or a long tube. The latter will be easier to work with.Sesame oil and seeds: This is the star ingredient! If your child is allergic to sesame, you can substitute with perilla oil and seeds, aka deulkkae. It has an earthy, slightly minty flavor. Mushrooms: You can choose any variety.Egg: If there's an egg allergy you can leave it out. But if not, HIGHLY recommended. It adds a lovely richness to the stew.Storage: Will keep in the fridge for 3-4 days.