Peel eggs and cut in half, either lengthwise or crosswise. Carefully scoop out yolks, and arrange whites on a platter or tray.

In the bowl of a food processor, process egg yolks and mayonnaise until smooth. Alternatively, press yolks through a sieve; mix yolks with mayonnaise until smooth. Season with salt, black pepper, and cayenne.

Transfer egg yolk mixture to a pastry bag fitted with a large plain or star tip. Pipe mixture carefully and very neatly into egg halves; top with assorted garnishes, if desired. Cover with plastic wrap, and refrigerate until ready to use, up to 3 hours.