Place eggs in a saucepan in a single layer and cover with cold water by 1.0 inch. Bring to a full boil over medium-high heat.
Once boiling, cover, turn off the heat, and let stand for 10 minutes. Transfer eggs to an ice bath until cool, about 5 minutes.
Gently tap and peel the eggs. Slice each egg lengthwise and remove yolks to a bowl; place whites on a serving platter.
Mash yolks with mayonnaise, Dijon, vinegar, salt, and pepper until smooth and creamy. Stir in pickle relish if using; adjust seasoning to taste.
Spoon or pipe the yolk mixture into the egg white halves, mounding slightly.
Dust with paprika and garnish with chives if desired. Chill until ready to serve.
Notes
For easier peeling, use eggs that are a few days old. Make up to 1 day ahead and refrigerate in an airtight container. Variations: swap Dijon for yellow mustard for a milder flavor, or add a dash of hot sauce for heat.