Put whites in a large bowl and let stand at room temperature 15 minutes. With an electric mixer beat whites until they hold stiff peaks and with a metal spoon thoroughly fold in Parmesan and cayenne. Roll mixture into about twenty 3/4-inch balls. In a 3-quart heavy kettle heat 1 inch oil to 375° F. Working in batches, fry balls, turning them, until golden, about 2 minutes, transferring to paper towels to drain. Serve Parmesan puffs warm as an accompaniment to soup or as an hors d’oeuvre.