In a large pot, bring salted water to a boil. Add the diced potatoes and cook for about 10-12 minutes or until tender. Add the diced carrots in the last 5 minutes of cooking. Drain and let cool.
In a large mixing bowl, combine the cooled potatoes and carrots with the diced cucumber and chopped green onions.
In a separate bowl, whisk together the mayonnaise, sugar, apple cider vinegar, salt, black pepper, garlic powder, and sesame oil (if using) until smooth.
Pour the dressing over the potato mixture and gently stir until everything is well coated.
If using, fold in the chopped hard-boiled eggs for added texture and flavor.
Cover the salad and refrigerate for at least 1 hour to allow the flavors to meld. Serve chilled.