Julienne the cucumber, carrot and red cabbage. Thinly slice the lettuce. Keep them cool in the fridge or in ice water (drain well before using) if you like.
Place the noodles in the middle of the serving bowl and nicely arrange the cold vegetables over the noodles. Drizzle the sesame oil over, sprinkle with the sesame seeds, and place a half of boiled egg on top. The sauce can be served on top or separately.
Alternatively, you can toss it all together before serving. Mix the noodles well with the sauce, add the sesame oil and vegetables, and toss well. Sprinkle with the sesame seeds and place a half of boiled egg on top.