I learned how to make this salad in elementary school in Japan. It's my go-to dish for taking to potlucks and barbecues. We took it to one Friday night, and it was kind of a hit.
4 big potatoes, peeled (white is good, but any kind is okay as long as they're big)
1-2 handfuls of baby carrots (depending on how much you like carrots)
1 cucumber, half-peeled (unpeeled is okay; "half-peeled" is peeling strips down the side so it looks like it's striped)
Kewpie Japanese mayonnaise (see illustration)
Fresh-ground pepper (to taste)
1. Boil the eggs.
2. Dice the potatoes into 1 inch cubes - not too small. Cut baby carrots into halves or thirds. Dump potatoes and carrots into large pot, fill with water, and bring to boil.
3. Boil potatoes and carrots until they're soft enough that you can easily pierce them with a knife, but not so soft that they crumble.
4. While you're waiting for the potatoes and carrots to boil, but the cucumber into thin slices. Sprinkle a little salt over the sliced cucumber. Then wrap the sliced, salted cucumber in a paper towel and squeeze. The idea is to get as much moisture out of the cucumber as you can, so that when you put them into the salad it doesn't get soggy.
5. Drain potatoes and carrots and return them to the pot you boiled them in. Keep the stove on high heat and hold the pot over the burner so it's not touching (if it's electric) or slightly high over the flame (if it's gas). Shake the pot so that the heat dries the potatoes and carrots completely, but don't let them burn. You'll know you're done when the sides and bottom of the pot turn whitish from potato starch, and when the corners of the potato cubes get rounded.
6. Dump potatoes and carrots into large mixing bowl. Squeeze some Kewpie over the top. You'll be using as much Kewpie as you feel you need - there's no set measurement to this. Add some, then mix, then add some more, mix some more, until you're satisfied. You may need to mash the potatoes and carrots a little while you mix, but don't mash them smooth. By squishing them a little, the taste of the potatoes and carrots mixes well with the Kewpie, but if you squish them too much the whole thing turns into a heavy paste. You want to keep it fluffy.
7. Using an egg slicer/dicer thingie, dice three of the boiled eggs and add to the mix.
8. Add cucumber and mix. Cucumbers should be added last, not long before eating, because they'll make the salad soggy. So, if you're making this in the morning to take to someone's barbecue in the evening, keep the cucumbers out and add them right before you leave in the evening.
9. Season to taste with pepper; mix.
10. Cut remaining boiled egg in half and remove the yolk. Crumble half the yolk (or all of it, depending on your preference) over the salad as a garnish. Voila!
This potato salad is better if it's not refrigerated, so if possible make it close enough to serving time that it's still a little bit warm.