Place the eggs in a large saucepan (with a tight fitting lid) in a single layer. Add cold tap water to cover by one inch. Heat on high until the water comes to a boil. While the water is heating, gently stir the eggs a few times to redistribute. Once the water comes to a boil, stir the eggs one more time, reduce the heat to LOW, cover the pan and cook for 13 minutes.
Remove the pan from the heat, and carefully drain the hot water. Cover the cooked eggs with cold water and stir. Drain then fill again with cold water. Allow the eggs to sit for 2-3 minutes.
Peel the eggs under cool water and place on a paper towel lined plate to dry. Slice each egg in half, and remove the yolk to the bowl of a food processor (if using. If not see NOTE) Place the now empty egg whites on a serving plate or in a container for storage. Set aside.
Add the mustard, mayonnaise, salt and pepper to the food processor with the egg yolks and pulse until smooth and blended. Remove the mixture to a small bowl and add the relish. Stir until blended. Check the seasoning and add more salt or pepper if needed. For an extra smooth filling add mayonnaise, one tablespoon at a time, until you reach the desired consistency.
Pipe or spoon the filling into the prepared egg whites. Garnish with paprika, cayenne or more black pepper if desired. Serve immediately or refrigerate until needed.