Place eggs in a single layer in a medium saucepan and cover with cold water by one inch. Bring to a full boil over medium-high heat, then remove from heat, cover, and let sit for 11 minutes.

While eggs sit, fill a large bowl with ice and cold water. Transfer eggs immediately to the ice bath and let them cool for at least 10 minutes. This stops the cooking and makes peeling much easier.

Peel the eggs, slice each one in half lengthwise, and pop the yolks into a mixing bowl. Set the whites on your serving platter, cut side up.

Mash the yolks with a fork until no large lumps remain, then stir in the mayonnaise, dijon mustard, vinegar, salt, and pepper. Mix until the filling is smooth and creamy. Taste and adjust salt if needed.

Spoon the filling into a piping bag or a zip-top bag with one corner snipped off. Pipe the filling into each egg white, mounding it slightly in the center.

Heat a small skillet over medium heat. No oil needed. Lay the prosciutto pieces flat in the pan and cook for 2 to 3 minutes per side until the edges curl and the meat turns deep pink and crispy. Transfer to a paper towel to cool.

Press one piece of crispy prosciutto on top of each filled egg. Dust lightly with smoked paprika and serve right away, or cover and refrigerate for up to 24 hours with the prosciutto added just before serving.