Place eggs in a single layer in a large saucepan and cover with cold water by 1 inch. Bring water to a rolling boil over high heat, then immediately remove from heat, cover, and let stand for exactly 12 minutes.
Transfer eggs to an ice bath and let cool for 15 minutes. This rapid cooling creates a slight separation between the shell and egg, making peeling easier.
Gently tap each egg on a hard surface and roll it to create cracks all over, then peel under a thin stream of cold water for the cleanest removal.
Cut the peeled eggs in half lengthwise. Carefully remove the yolks and place them in a medium bowl. Arrange the whites on a serving platter or deviled egg tray.
Using a fork, mash the yolks until they reach a fine, powdery consistency to ensure your filling will be silky smooth rather than lumpy.
Add mayonnaise, mustard, pickle relish, vinegar, salt, pepper, and paprika to the mashed yolks. Mix until completely smooth and creamy.
Spoon or pipe the filling into the egg white halves. For a professional presentation, use a piping bag fitted with a star tip.
Sprinkle with additional paprika and garnish with minced chives if desired. Refrigerate until ready to serve.