Place eggs in a single layer in a saucepan and cover with cold water by an inch. Bring to a boil, then cover, remove from heat, and let sit for 11 minutes.
Transfer the eggs to an ice bath for 10 minutes, then peel. Store peeled whole eggs in an airtight container in the fridge for up to 2 days if not serving right away.
When ready to make the filling, halve the eggs and scoop the yolks into a bowl, keeping the whites separate.
Mash the yolks with mayonnaise, Dijon mustard, vinegar or pickle brine, salt, and pepper until smooth.
Store the filling in an airtight container or zip-top bag, and store unfilled whites separately, cut-side down on a paper towel, for up to 4 hours before serving.
About an hour before serving, pat the whites dry again, then pipe or spoon the filling into each half.
Sprinkle with paprika and serve chilled.