Creating New York Deli Deviled Eggs is a straightforward process. Follow these steps to achieve deviled egg perfection:

1. Boil the Eggs:
   Place the eggs in a large saucepan and cover them with cold water. Bring the water to a boil over medium-high heat. Once boiling, cover the saucepan, remove it from the heat, and let the eggs sit for 12 minutes.
2. Cool and Peel the Eggs:
   After 12 minutes, transfer the eggs to a bowl of ice water to cool for at least 5 minutes. Once cooled, gently peel the eggs and slice them in half lengthwise.
3. Prepare the Filling:
   Carefully remove the yolks from the egg halves and place them in a mixing bowl. Set the egg whites aside. Mash the yolks with a fork until they have a fine crumbly texture.
4. Mix the Ingredients:
   Add mayonnaise, yellow mustard, horseradish, dill pickle relish, red onion, capers, dill, apple cider vinegar, salt, and pepper to the mashed yolks. Mix until the ingredients are well combined and the mixture is creamy and smooth.
5. Fill the Egg Whites:
   Use a spoon or a piping bag to fill the hollowed-out egg whites with the creamy yolk mixture. Ensure that each egg half is generously filled.
6. Garnish and Serve:
   Sprinkle smoked paprika over the filled eggs for a touch of color and flavor. Finish with a garnish of freshly chopped chives.