Cook the noodles according to package directions. Drain and rinse under cold water. Drain again, and toss with the sesame oil. Transfer the noodles to a large serving bowl.
Arrange the cucumber in a ring along the outer edge of the bowl. Make another ring inside of the cucumber with the carrots. Place the bean sprouts in the center, on top of the noodles. Set aside.
For the crepes: Heat a wok over high heat. Lightly beat the eggs and add the salt. Once the wok is hot, add the peanut oil and swirl to coat the pan. Pour the eggs into the center of the wok and tilt the pan so the eggs comes partly up the sides. Keeping the pan over the heat, swirl until the eggs are set. Roll the eggs into an omelette shape and remove from the pan. Slice the omelette into strips and lay over the vegetables and noodles. Sprinkle with the shichimi togarashi.
For the dressing: Using the side of a knife, smash the garlic into a paste. Transfer the garlic to a small mixing bowl. Add the sesame paste, soy sauce, vinegar and chile sauce. Stir to combine. Loosen with enough water to make the sauce thin enough to pour.
Serve the noodles at room temperature with the dressing on the side.