Place the eggs in a single layer in a saucepan and cover with enough water to completely cover the eggs. Heat on high until water begins to bubble, then bring to a simmer and cover. Cook for 10 minutes. Remove from heat immediately submerge eggs in ice water to cool.
Crack egg shells and carefully peel under cool running water. Gently dry the eggs and refrigerate in airtight container if making the next day. Otherwise, proceed to next step.
Slice the eggs in half lengthwise and remove the yolks to a bowl. Mash the yolks and add in the mayonnaise, mustard, vinegar, and salt. Mix well.
Set the egg whites on a tray, cut side up. At this point you could either spoon the yolk mixture evenly into all of the eggs, or pipe the yolks in to create a fancier look. To do this, put the yolk mixture into a ziplock bag, snip the corner of the bag and pipe into the eggs. Sprinkle with paprika and serve.
Notes
You could also garnish this with chopped dill, sliced green onions, or chopped shallots.