Place the 6 eggs in a saucepan and cover them with cold water by about 1 inch. Bring the water to a boil, then cover the pan, remove it from the heat, and let the eggs stand for 10-12 minutes.

Transfer the cooked eggs immediately to an ice-water bath and let them chill for at least 5 minutes. The rapid cooling stops the cooking process and helps loosen the shells.

Peel the eggs carefully, rinse away any shell fragments, and pat them dry. Slice each egg lengthwise and gently remove the yolks, placing them in a mixing bowl.

Mash the yolks with the mayonnaise, mustard, vinegar or pickle juice, salt, and black pepper until smooth. Stir in the relish if using, and add a little more mayonnaise only if the mixture needs a softer texture.

Spoon or pipe the yolk mixture into the egg-white halves. Finish each half with a light dusting of paprika, then chill the assembled deviled eggs before serving.