Peel and rinse the potatoes. Slice the potatoes into ¼-inch thick slices and put them into a medium pot. Add enough water to just cover the potatoes. Cover the pot and bring it to a boil over medium-high heat. Cook for the potatoes for 18 minutes.
While the potatoes are cooking, peel and rinse the carrots. Cut the carrots into ¼-inch slices. Add the carrots to a small pot, and cover with about 1 inch of water. Cover the pot and bring to a simmer, then cook, uncovered, for about 3 to 5 minutes, or until the carrots are fork-tender. When the carrots are done cooking, drain and add to a large mixing bowl.
Rinse and finely chop the cucumber and set aside.
Mince the onion. Add the apple cider vinegar, salt, and ½ teaspoon sugar to the minced red onion and stir with a spoon. Set aside.
When the potatoes are done cooking, drain them. Use a ricer to get the potatoes to a very fine mashed texture in a large mixing bowl.
Cut two of the eggs in half and remove the yolks. Press the yolks through a mesh sieve to finely grate the yolks and set aside.
Chop the remaining 2 eggs and 2 egg whites and add them to the potatoes and carrots.
Add the red onion and vinegar mixture to the potatoes and carrots. Add the mayonnaise and the cucumbers. Mix everything together with a spatula.
Serve the potato salad with an ice cream scoop and top with the grated egg yolks.