Place eggs in a single layer in a saucepan and cover with cold water by 1 inch. Bring to a rolling boil over medium-high heat. Once boiling, remove from heat, cover, and let stand for exactly 12 minutes.
Transfer immediately to an ice bath to stop the cooking process. Cool for at least 15 minutes.
Gently tap each egg on a hard surface and roll between your palms to loosen the shell. Begin peeling from the wider end where the air pocket is located. Rinse under cool water to remove any shell fragments.
Halve each egg lengthwise and gently remove the yolks into a medium bowl. Arrange the whites on a serving platter.
Use a fork to mash the yolks until completely crumbled and no large pieces remain.
Add mayonnaise, mustard, vinegar, salt, and pepper to the yolks. Mix until perfectly smooth and creamy. If using, add the pickle relish and mix well.
Transfer the filling to a piping bag fitted with a star tip (or use a zip-top bag with the corner cut off for a simpler approach). Pipe the filling into each egg white half, creating an elegant swirl.
Sprinkle with paprika for that classic look and add chopped chives for a fresh color contrast and subtle onion flavor.