Cut the hard-boiled Irvine’s Eggs in half. Arrange the egg white shells cut side up on a serving plate and put the yolks in a small mixing bowl.
Mash the yolks with a fork then stir in mayonnaise, mustard, and vinegar. Mash and stir all the ingredients together until well combined and the egg mixture is smooth. Taste, and add salt and pepper if desired.
Pipe a little bit of the mixture into each egg white shell, dividing the mixture as evenly as possible between the eggs. Sprinkle the eggs with paprika if desired.
Top with olives and serve immediately or refrigerate until needed.