Make the mashed potatoes by placing the potato cubes in a saucepan and cover with water. Bring to a boil over high heat and time for 15 minutes. Lower heat and simmer until tender and the potatoes just break apart when you pierce them with a steak knife. (You do not want the potatoes to be mushy.)
Drain and transfer the potatoes to a mixing bowl. Add the butter, water, cheese, and chives. Mix on medium speed to combine.
Add the egg and beat just until smooth.
Add the flour and combine until smooth. Season with salt and pepper to taste.
Preheat the oil to 375ºF.
While the oil heats, roll the dough into 1-inch balls (a little less than one tablespoon each); set aside. Set out two plates lined with three layers of paper towels; set aside.
Deep-fry the balls for 5 minutes or until they become golden brown. Allow the excess oil to drip back into the fryer and transfer the croquettes to the prepared plates.
Serve warm or at room temperature with your desired sauce(s).
Tips
Variation:
Add some spices to the dough such as cayenne pepper or chili powder and cumin for a Mexican flare.