In a large mixing bowl, combine the beaten eggs and oil, then add the cabbage, carrots, sliced green onions, and sourdough starter. Mix them well until they are thoroughly combined. 

The batter should be thick but spreadable, like sourdough pancake batter. If it's too thick, add water 1 tablespoon at a time until it reaches the right consistency. This is a great way to use extra sourdough starter.

Heat a nonstick skillet, cast-iron skillet, or hot griddle over medium-low heat (about 325°F / 165°C).

Lightly oil the surface, then scoop about  1 1/2 cup of batter into the hot pan.

Spread it into a round pancake shape about 1/2-inch thick.

Cook on the first side for 3–4 minutes, until the bottom is golden brown and the edges are set. 

Using a large spatula (or two spatulas for support), flip it carefully and cook on the second side for another 3–4 minutes, until both sides are crisp and the pancake is cooked through.

Repeat this process with the remaining batter, lightly oiling the pan between each batch for best results.

Transfer the warm okonomiyaki to serving plates and top them with a generous spread of okonomiyaki sauce and Japanese mayo. Garnish it with the reserved green onions and your favorite pancake toppings. Serve them hot and enjoy this delicious breakfast!