Rinse the sushi rice in several changes of water until the water runs nearly clear, then drain.
Combine about 525–550 ml water and a pinch of salt in a large pot and bring to a boil; add the rice, cover, and simmer over low heat until the water is absorbed and rice begins to stick to the pot, about 12–15 minutes. Do not stir frequently.
Heat the rice vinegar with the sugar and 1 tsp salt until the sugar dissolves; set aside to cool.
Transfer the cooked rice to a large bowl, pour the vinegar mixture over it, and gently fold to combine while fanning or letting it cool to room temperature; keep the rice slightly warm but not hot.
Cut the chicken breast into thin strips and season with salt and pepper.
Dredge the chicken strips in flour, then dip into a batter made from the beaten egg, 1 tbsp flour, and enough cold water to make a thin batter; fry in hot oil in a pan until golden and cooked through. Drain on paper towels.
Make the omelet: beat 4 eggs with 1 tsp soy sauce and the chopped green onion. Cook in a non-stick pan with a little butter over medium heat, rolling or folding as it cooks; when set, remove and slice into ~1 cm strips after cooling slightly.
Prepare the vegetables: julienne the cucumber, carrots, and thinly slice the bell pepper; finely chop the lettuce.
Place a nori sheet shiny-side down on a bamboo rolling mat lined with foil (optional). Wet your hands with cold water to prevent sticking, then spread a thin, even layer of rice over about 3/4 of the nori.
Arrange strips of fried chicken, omelet, cucumber, carrot, bell pepper, and chopped lettuce across the rice near the edge closest to you.
Use the mat to roll the nori tightly over the filling, pressing the seam and ends to seal; keep the roll in the mat briefly and press the ends well.
With a sharp knife wiped with a damp cloth between slices, cut the roll into pieces about 1–1.5 cm thick.
Arrange the kimbap pieces on a platter and serve with soy sauce, pickled ginger, or wasabi if desired.