Place the eggs in a single layer in a large pot and cover with cold water by about an inch. Bring to a boil over medium-high heat. Once boiling, cover, remove from heat, and let sit for 12 minutes.

While the eggs cook, prepare an ice bath in a large bowl. When the 12 minutes are up, transfer the eggs immediately to the ice bath. Let them cool completely, about 15 minutes.

Peel the eggs carefully and slice each one in half lengthwise. Gently scoop the yolks into a medium bowl. Arrange the whites on a serving plate.

Mash the yolks with a fork until smooth. Add the mayonnaise, mustard, and vinegar. Mix until light and fluffy. Season with salt and pepper to taste. Stir in the pickle relish if using.

Spoon or pipe the filling into each egg white half, dividing evenly among the 24 halves.

Sprinkle with paprika just before serving. Keep refrigerated until ready to serve.