Boil the eggs: Place the eggs in a medium (2-3 quart) saucepan and fill with enough water to cover the eggs by 2 inches. Cover and bring to a boil, then remove from the heat and let rest, still covered, for 8 minutes.
Fill a large bowl halfway with ice water.
Drain the eggs and shake gently in the pan to crack their shells.
Place the eggs in the ice water bath and let sit for 5 minutes before peeling.
Peel the eggs, then cut the eggs in half and gently scoop the yolks into a mixing bowl.
Make the filling: Mash the yolks with a fork, then stir in the mayonnaise, relish, mustard, salt and pepper. Transfer the yolk filling to a zip-top bag, seal shut, and refrigerate.
Place the egg whites in an airtight container and refrigerate.
To assemble: Place the egg whites on a serving platter. Snip a corner off the still-sealed zip-top bag and use the bag as a piping tool to fill the egg whites.
Sprinkle with paprika and serve.