Wash and peel the potatoes and cut them into equal sized chunks if they are large.
Cover them with cold water and bring to a boil.
Add salt and boil until a butter knife pierces through easily.
Strain all the water and pass through a potato ricer (preferred method) or mash until just mashed -- you don't want to make them dense here (which is why the potato ricer is worth investing in!). Let them cool.
To make the croquette filling, mix the riced potatoes with the beaten egg, salt, pepper, and cheese. Do not overmix. If you want to taste the filling for seasoning, just add the egg as the last step.
Next, while heating a pan full of olive oil on the stove, pass the croquettes through the three step breading process. First, cover them in flour, then in beaten egg, and, finally, in the breadcrumbs.
Deep fry the croquettes in hot olive oil (if available) or vegetable oil/sunflower oil if not. They should take 3-4 minutes to brown. 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 rest on paper towels and serve hot with dipping sauces if desired.