Shred cabbage as finely as possible into thin ribbons.
Whisk eggs, water, soy sauce, sesame oil, salt, and pepper together. Add flour and whisk until smooth. Fold in shredded cabbage and green onions until fully coated. The mixture should be thick with cabbage.
Stir together all okonomiyaki sauce ingredients in a small bowl. Set aside.
Heat 1 tablespoon oil in a large non-stick skillet over medium heat. Scoop about 1 cup of batter per pancake into the pan and press firmly into a flat disc about ¾-inch thick. Cook undisturbed for 4 to 5 minutes until the bottom is deeply golden.
Slide a wide spatula fully under the pancake and flip in one confident motion. Press flat again. Cook 3 to 4 more minutes until golden on the second side and cooked through.
Transfer to a rack. Repeat with remaining batter, adding oil between pancakes.
While hot, drizzle okonomiyaki sauce over each pancake, then zigzag Japanese mayo over the top. Scatter sesame seeds, green onions, and bonito flakes. Serve immediately.