Place eggs in a single layer in a saucepan and cover with cold water by at least one inch. Bring water to a rolling boil, then immediately remove from heat, cover, and let stand for exactly 12 minutes.
Immediately transfer eggs to an ice bath and let cool for 5 minutes. This quick-cooling technique makes peeling significantly easier.
Gently tap eggs on countertop and roll to crack shell all over. Peel under cool running water to help remove any stubborn shell fragments. Chop eggs into your preferred consistency.
In a medium bowl, combine mayonnaise, mustard, relish, vinegar, celery salt, paprika, white pepper, and that pinch of sugar. Whisk until smooth and creamy.
Fold in the finely minced onion and celery.
Add chopped eggs to the dressing mixture and fold gently with a rubber spatula. Season with salt to taste.
Cover and refrigerate for at least 30 minutes before serving to allow flavors to meld and develop.