Step 1

In a medium bowl, stir together the cucumber, carrot, onion and 2 teaspoons salt; set aside.In a large saucepan over high, combine the potatoes with enough water to cover by 1 inch. Add 1 teaspoon of salt and bring to a boil. Reduce heat to medium-high and simmer until a skewer inserted into the potatoes meets no resistance, 12 to 15 minutes.

Step 2

Drain the potatoes, then transfer to a large bowl. Using a fork, coarsely mash half of the potatoes. Sprinkle with the vinegar and ¾ teaspoon of pepper.Stir to combine, then spread in an even layer along the bottom and sides of the bowl. Let cool for at least 20 minutes.

Step 3

Transfer the vegetable mixture to a fine mesh strainer, rinse well and drain. Working in batches, use your hands to squeeze the vegetables, removing as much liquid as possible, then add to the potatoes.Add the mayonnaise, ham, whole egg and yolk and the sugar. Fold until well combined. Taste and season with salt and pepper, then sprinkle with scallions. Serve at room temperature or chilled. Fold until well combined. Taste and season with salt and pepper, then sprinkle with scallions. Serve at room temperature or chilled.