Place the eggs in a medium saucepan. Cover with cold water by at least 1 inch and bring to a boil over high heat. Decrease the heat to just below simmering. Cover the pan and cook for 15 minutes. Immediately run cold water over the eggs: to quickly cool, add ice cubes. Let stand 2 minutes. Drain. Crack them on the countertop and slide the shells off.

Halve the eggs lengthwise. Remove the yolks to a small bowl; place the whites on a platter. If you are not filling the eggs immediately, cover and refrigerate until ready to prepare.

Mash the yolks with a fork. Add the mayonnaise, vinegar, mustard, butter, salt, and pepper. Mix well. Fill the whites evenly with the yolk mixture. Sprinkle with paprika. Cover and refrigerate until ready to serve.