Special equipment: a deep-frying thermometer; a pastry bag fitted with a large open star tip
Pour oil into a large pot to a depth of 2 inches. Heat over medium heat until the oil registers 340 to 350 degrees F on a deep-frying thermometer.
In a saucepan, combine the milk, water, butter and a generous pinch of salt. Bring to a simmer over medium heat, allowing the butter to melt completely. Add the cayenne to the flour and, all at once, pour the mixture into the saucepan. Cook, stirring with a wooden spoon, until the "paste" pulls away from the pan, about 2 minutes. Continue stirring for another 2 minutes to allow the moisture to evaporate.
Off the heat, add the mashed potatoes to the saucepan. Return to the heat and stir for another minute. Taste and adjust the seasoning with salt and pepper if needed.
Remove from the heat and continue stirring. Whisk the eggs together and, while stirring, add the eggs to the mashed potato mixture. Continue stirring until a smooth paste is formed. Stir in the Parmesan and season with pepper.
Transfer the mixture to a large pastry bag fitted with a large open star tip. In batches, pipe inch-long dollops of the choux paste into the hot oil, using a knife or the side of the pan to cut the paste. Fry until the fritters are deep golden brown all over, 3 to 4 minutes. Remove with a slotted spoon and turn out onto a paper towel-lined plate.
If you don't have leftover mashed potatoes, substitute 1 large Yukon Gold potato, peeled, cooked to fork-tender and riced.