Peel the eggs and cut in half, either lengthwise or crosswise. Carefully scoop out the yolks, and arrange the whites on a platter or tray. (You can set the eggs on a thick bed of dill, parsley, or other green to keep them from rolling)
In the bowl of an electric mixer or food processor, blend the egg yolks, mayonnaise, and melted butter until smooth. Season to taste.
Put the egg yolk mixture in a pastry bag fitted with a large star tip. Pipe the mixture carefully and very neatly into the egg halves. Cover with plastic wrap and refrigerate until ready to use, up to 3 hours.
Two tablespoons Dijon or yellow mustard and or 1 tablespoon curry powder, 1 to 2 tablespoons chopped fresh dill or chervil or 2 tablespoons very finely chopped ham can be added to the blended yolk mixture.
Deviled eggs can be topped with spoonful of red, back or golden caviar.