Wash and peel the potatoes and cut them into equal sized chunks if they are large, then place them in a large pot and cover them with cold water. Add salt and boil until a butter knife pierces through the potatoes easily, about 20-25 minutes.
Drain the potatoes, then pass them through a potato ricer and let them cool. You can use a potato masher or stand mixer to mash them until smooth, but don't overdo it, or they may be dense--which is why the potato ricer is the best option!
To make the croquette filling, mix the riced potatoes with the salt, pepper, and cheese, then add more seasonings to taste. Add the egg last, being careful not to overmix.
Roll the croquette mixture into balls, then coat them in flour, then beaten egg, then in the breadcrumbs to coat.
Meanwhile, heat a pan of olive oil on the stove over medium heat.
Deep fry the croquettes in the hot oil, turning as they cook; about 3-4 minutes. You can also pan fry them in less oil, flipping halfway through, but make sure they cook evenly on all sides.
Let the potato croquettes drain on paper towels, then serve them hot with dipping sauces if desired. Enjoy!