Cook the pasta al dente according to the package directions. Rinse under cold water and drain.
Slice the onion thinly and sprinkle with a generous pinch of salt. Massage the salted onions and let it sit for about 10 minutes. Rinse with water then squeeze the washed onions to get as much moisture out.
Cut up ham into small strips. Seed and partially peel the cucumbers (leaving some green peel for color) then slice them into matchsticks.
In a large bowl, whisk together all the ingredients for the dressing. Add pasta, cucumbers, ham and onions into the dressing. Mix to coat the dressing.
Garnish with hard boiled eggs and green onion.
Cover and refrigerate at least 1 hour before serving. (Optional: sprinkle Shichimi or paprika pepper right before serving.)