Place the eggs in a large saucepan and fill the pan with cold water until the eggs are completely submerged. Add the vinegar and 1 teaspoon of salt.
Heat the saucepan over medium-high heat and bring the water to a boil. Remove the pan from the heat and cover it with a lid. Allow the eggs to sit in the hot water for 15 minutes.
As the eggs near the end of their cooking time, fill a large bowl with ice water.
Using a slotted spoon, remove the eggs, one by one, from the pan. Tap each egg lightly on the counter to crack the shell.
Place each egg in the bowl of ice water to prevent further cooking. Once the eggs are thoroughly cooled, begin peeling them under cool running water.
Carefully slice each egg in half lengthwise. Scoop out each yolk. Place the yolks in a large bowl.
Using a fork, mash the yolks with the mayonnaise. Add the relish, mustard, pepper, and remaining salt; combine.
Spoon the egg mixture into each halved egg. To pipe the mixture into the halved eggs, you can use a disposable piping bag or a Ziploc bag with one corner cut off.
Garnish each egg with a dash of paprika. Enjoy!
See this article to learn how to peel hard-boiled eggs like a chef.