Prepare the Cabbage and Veggies: Finely shred the cabbage and grate the carrot if using. Chop the scallions.
Make the Batter: In a large bowl, whisk together the eggs, flour, water, salt, and pepper until smooth.
Combine Ingredients: Add the shredded cabbage, scallions, and grated carrot to the batter. Mix well until all the veggies are evenly coated.
Heat the Pan: Preheat a non-stick skillet over medium heat and add 1 tablespoon of oil.
Cook the Pancakes: Spoon about 1/4 cup of the mixture onto the skillet, flattening gently with the back of the spoon to form a pancake. Cook for 3–4 minutes on each side, or until golden brown and crispy.
Repeat: Add more oil as needed and continue cooking the remaining batter.
Serve: Serve warm with soy sauce, a dipping sauce, or your favorite condiment.