Peel and cut the potatoes into even chunks. Thinly slice the carrot and cucumber. Salt the cucumber, let it sit a few minutes, then squeeze out the water.
Boil the potatoes until tender, about 12-15 minutes, adding the carrot for the last few minutes. Drain well.
Prepare the ham: cut into pieces and add as-is, or lightly fry to deepen the flavor. (No ham? Bacon works as a substitute — fry it until browned.)
Mash the potatoes while hot, leaving some lumps. Let cool slightly.
Add the carrot, squeezed cucumber, and ham. Mix in the Kewpie mayonnaise, salt, and pepper until creamy. Taste, adjust, and serve.
