In a bowl, whisk the eggs until well combined. Add the soy sauce or tamari, season with about 1 teaspoon of sea salt and a few pinches of white pepper and whisk well.
Drain the wakame and squeeze out the excess water. If your wakame is in large chunks, roughly chop. If they are smaller pieces, leave them as is.
Heat a large frying pan over medium–high heat. When hot, drizzle with olive oil, add the wakame and stir for 1–2 minutes, until the liquid has evaporated and the wakame starts to char slightly. Pour the egg into the pan, allow to set for 10 seconds, then use a rubber spatula to push the egg around the edge of the pan in a circular fashion, tilting the pan to let the runny egg in the middle run out to the edge. Repeat until the egg is just set with a slightly wet centre – this should only take 20–30 seconds. Remove from the heat immediately and transfer to bowls.
Top the scramble with the green onion, a drizzle of sesame oil and the sesame seeds, and serve with white rice.