Place eggs in a single layer in a saucepan and cover with cold water, ensuring there's an inch of water above the eggs. Bring to a rolling boil over medium-high heat.
Once boiling, remove the pan from heat, cover, and let stand for exactly 12 minutes.
Immediately transfer the eggs to an ice water bath and let them cool for 15 minutes.
Gently tap each egg on a hard surface, creating cracks all around. Start peeling from the wider end where the air pocket is located, and peel under a thin stream of cold running water.
Slice each peeled egg lengthwise with a sharp knife. Carefully remove the yolks and place them in a medium bowl. Arrange the whites on a serving platter or deviled egg tray.
Mash the yolks with a fork until no large pieces remain. Add mayonnaise, mustard, vinegar, salt, and pepper (and optional pickle relish if using). Mix until smooth and creamy.
Transfer the yolk mixture to a piping bag fitted with a star tip, or use a resealable plastic bag with the corner snipped off. Pipe the filling into each egg white half, creating an attractive swirl pattern.
Sprinkle with paprika for the classic finish. For a more impressive presentation, add a small dill sprig, sliced olive, or tiny piece of crisp bacon to each egg.
Refrigerate until ready to serve, ideally within 2 hours of assembly for optimal flavor and food safety.