Rinse the sushi rice under cold water several times until the water runs clear; drain.
In a pot, bring 525–550 ml water to a boil with a pinch of salt, add the rice, cover, and simmer over low heat until the water is absorbed and rice begins to stick to the pot; avoid stirring.
Heat the rice vinegar with the sugar and 1 tsp salt until the sugar dissolves; transfer cooked rice to a large bowl and gently fold in the vinegar mixture to season and cool the rice.
Prepare the chicken by cutting into thin strips, season with salt and pepper, dredge in flour, then dip into a batter of the beaten egg, 1 tbsp flour, and enough cold water to reach a thin batter consistency.
Fry the battered chicken strips in hot oil in a frying pan until golden brown and cooked through; drain on paper towels and set aside.
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, roll as it cooks, then slice into ~1 cm strips after cooling.
Prepare the vegetables: julienne the cucumber and carrots, thinly slice the bell pepper, and finely chop the lettuce greens.
Place a nori sheet on a bamboo mat (covered with foil if desired). Wet your hands with cold water, spread a thin, even layer of rice over about three-quarters of the nori.
Arrange a line of fried chicken strips, omelet strips, cucumber, carrot, bell pepper, and chopped lettuce across the rice near the bottom third of the sheet.
Use the mat to roll the nori tightly over the fillings, pressing the ends to seal; keep the roll in the mat and press to firm it up.
With a sharp knife wiped on a damp cloth between cuts, slice the roll into pieces about 1–1.5 cm thick.
Arrange the kimbap pieces on a platter and serve (optional accompaniments: soy sauce, pickled ginger, wasabi).
