Cook the spaghettini al dente. Drain and rinse under cold water. Drain very well again. With your hands, toss the noodles with half of the sesame oil. Cover and chill until ready to combine with the peanut sauce, or up to 24 hours.
To make the peanut sauce, combine the remaining sesame oil with all the other ingredients except the cucumber and scallion. Beat until well mixed. The sauce can be kept in the refrigerator, covered, for up to 24 hours before serving.
Just before serving, gently toss the noodles with the sauce, cucumbers and half the scallions. Garnish with the remaining scallions.