When ready to cook, add ice to a large bowl of cold water and set aside.
Bring a large pot (at least 3 quarts) of water to boil. Use a spider or slotted spoon to gently lower the eggs into the pot and continue to boil for 30 seconds. Cover tightly, reduce heat to very low, a bare simmer, and continue cooking for 13 minutes. Remove the eggs to the bowl of ice water and allow them to cool for at least 15 minutes before peeling under cool running water.
Cut the eggs on half. Remove the yolks and place them in the bowl of a food processor (or mash by hand in a medium mixing bowl). Add the mayonnaise, mustard, vinegar, salt and pepper and process until smooth, scraping down the sides of the bowl as necessary, about 1 minute. Add the relish and chives and pulse for a second or two until everything is fully combined.
Transfer the yolk mixture to a piping bag with a star tip, or a sealable plastic bag*. Snip off a corner of the bag and pipe the filling mixture into egg whites, slightly overstuffing each. You may have one or two extra whites (save them for another recipe). Garnish each egg with a sprinkle of paprika or chili flakes, a few minced chives and/or a sprig of dill, and a little dollop of salmon roe, if you're feeling fancy,
Serve immediately or cover loosely with plastic wrap and refrigerate for up to 3 hours.