Peel potatoes and cut into ½-inch cubes. Immediately place them in a large pot with water to cover; season with salt. Bring to a boil over high heat and then decrease the heat to low. Simmer until tender, about 30 minutes. Drain well in a colander. While still warm, transfer the potatoes to a baking sheet and drizzle with vinegar. Set aside to cool to room temperature.
Halve 2 of the eggs lengthwise and remove the yolks (set aside for another use or discard). Grate the remaining egg whites and 3 whole eggs. Place in a medium bowl. Chef’s note: To easily peel hard-boiled eggs, gently tap shells on the counter to crackle the egg. Roll between your hands to loosen the shell. Peel, starting at the large end, while holding the egg under running cold water; this facilitates peeling and also removes any stray shell fragments.
To make the dressing, combine the mayonnaise, yogurt, grated eggs, onion, celery and pickle relish in a bowl. Once the potatoes have cooled completely, using a slotted spoon to leave the vinegar behind, transfer the potatoes to a large bowl. Pour the dressing over the potatoes and stir to combine. Taste and adjust for seasoning with salt and pepper. Serve at room temperature or chilled.