Stir the cucumbers and ½ teaspoon salt in a medium bowl. Let sit until the cucumbers release their liquid, about 30 minutes. Drain and place them on a paper towel-lined plate to dry.

Meanwhile, bring a large pot of salted water to a boil over high heat. Add the carrots and cook until bright orange and crisp-tender, about 2 minutes. Remove the carrots from boiling water using a slotted spoon or a spider and immediately place them in a large bowl of ice water to stop them from cooking. Scoop the carrots out of the ice bath and transfer them to the paper towel-lined plate with the cucumbers. Reserve the ice bath.

Bring the water in the pot back to a boil over high heat. Gently add the potatoes and eggs. Boil the eggs for 10 minutes. Transfer the boiled eggs to the reserved ice bath (adding more ice if needed). Continue to cook the potatoes until tender and a fork can be inserted without resistance, another 6 to 8 minutes. Drain them and return them to the pot. Shake the pot over medium heat to dry the potatoes. Remove from the heat and set aside.

While the potatoes finish cooking, peel the eggs. Separate the egg whites from the egg yolks. Roughly chop the egg whites and transfer to a medium bowl. Finely crumble the egg yolks and transfer all but 1 tablespoon of the yolk to the same bowl. Place the reserved 1 tablespoon egg yolk in a small bowl, cover, and refrigerate until ready to serve.

Whisk the mayonnaise, vinegar, sugar, and remaining ½ teaspoon salt in a large bowl until combined and the sugar dissolves.

Add the potatoes and lightly mash with a potato masher until the dressing is incorporated. Fold the vegetables and eggs into the potatoes until just incorporated. Cover and chill for at least 1 hour or up to 3 days. Season with salt to taste and garnish with the reserved crumbled egg yolk before serving. Love the recipe? Leave us stars and a comment below!