Hard boil the eggs:
Place eggs in a pot and cover with water. Bring to a boil, then turn off the heat, cover, and let sit for 10–12 minutes. Drain and transfer to an ice bath. Peel once cooled.

Halve the eggs:
Slice eggs in half lengthwise. Carefully remove yolks and place in a bowl.

Make the filling:
Mash the yolks with mayonnaise, mustard, salt, and pepper until smooth. Add vinegar or hot sauce if using.

Fill the eggs:
Spoon or pipe the yolk mixture back into the egg whites.

Garnish:
Sprinkle with paprika for a classic touch.

Serve chilled or at room temperature.