Drain kimchi thoroughly through a sieve and reserve the juice. Chop kimchi finely.
Combine egg, 3 tablespoons kimchi juice, water, brown sugar, salt, and sesame oil in a large bowl. Whisk together. Add green onions and kimchi. Stir in flour just until combined.
Heat oil in a skillet over medium-high heat. Add a few spoonfuls of pancake batter; any size pancake will do. Cook until edges are lightly browned, about 3 minutes. Flip and cook until crispy, 2 to 3 minutes more. Transfer pancake to a serving plate and repeat with remaining batter. Sprinkle bonito flakes on top.