Boil the potatoes with the skins on until cooked, then peel and mash them.
Thinly slice the cucumber and carrot, sprinkle with salt, and drain the excess water.
Chop the hard-boiled egg.
Combine the mashed potatoes, cucumber, carrot, egg, mayonnaise, salt, and pepper, and mix well.
Fill the morning bun with the potato salad to finish.