Bring potatoes, 1 Tbsp. salt, and enough water to cover by 1 inch to boil in large pot over high heat. Reduce heat to medium and simmer until potatoes are just tender, about 10 minutes.
Microwave vinegar and sugar in small bowl until sugar dissolved, about 30 seconds. Process vinegar mixture, mustard 1 hard-cooked egg yolk (reserve white), celery seed and 1/2 tsp. salt in food processor until smooth. Transfer to medium bowl.
Drain potatoes thoroughly and transfer to large bowl. Drizzle 2 Tbsp. dressing over hot potatoes and using a rubber spatula, gently toss until evenly coated. Refrigerate until cooled, about 30 minutes.
Whisk mayonnaise into remaining dressing. Add remaining egg white and 3 hard-cooked eggs to dressing and using potato masher, mash until only small pieces remain. Add dressing and celery to cooled potatoes, tossing to combine. Cover and refrigerate until chilled, about 30 minutes. Season with salt and pepper to taste.