In a large mixing bowl, combine the flour, water, salt, and sesame oil. Whisk until the batter is smooth and free of lumps.
Stir in the chopped green onions and grated carrot (if using) until evenly distributed.
Heat a non-stick skillet over medium heat and add 1 tablespoon of vegetable oil.
Pour half of the batter into the skillet, spreading it evenly to form a pancake about ½ inch thick.
Cook for 3-4 minutes until the bottom is golden brown and crispy. Flip the pancake and cook for another 3-4 minutes on the other side.
Remove the pancake from the skillet and place it on a paper towel to absorb excess oil. Repeat with the remaining batter.
Cut the pancakes into wedges and serve hot with soy sauce or a dipping sauce of your choice.