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.
