Arrange eggs in a saucepan and cover with cold water by about 1 inch.
Bring to a boil over medium-high heat. Cover, remove from heat, and let sit 12 minutes.
Transfer eggs to an ice bath for 5 minutes. Tap and peel under cool running water.
Slice eggs lengthwise. Gently pop yolks into a bowl and set whites on a platter.
Mash yolks with mayonnaise, Dijon, vinegar, salt, and pepper until smooth. Stir in relish and chives, if using. Add a splash of water to loosen if needed.
Spoon or pipe the filling into egg whites. Dust with paprika and chill until serving.