The last say of sailing in the BVI I had a bag of potatoes, an onion, and a half-pound of cheese left, and we were returning the boat by 11am. Therefore, we had cheesy potatoes for breakfast. It goes like this:
I was working with small red potatoes (often called "creamers"). Boil the potatoes in water for about 15 minutes until you can get a skewer through them kind of easily, but not really easily. You want them "par-boiled" or almost-cooked. Drain the water and cut them into quarters.
During that time that the potatoes are boiling, slice your cheddar cheese and your onion.
Have a (fairly large) skillet on medium-high heat. Add olive oil to coat. Don't go crazy -- we're not deep-frying -- but don't be shy either: you want a good coating on the entire surface of the pan, plus a little bit more. Wait about a minute to let the oil heat up. If you tilt the pan, it should run quickly and be much thinner than it was in the bottle. Butter also works well, either alone or in addition to the oil.
Add the onion. Move it around for a few minutes so it is translucent and just starting to brown in a few places.
Add the potatoes in with the onions. Move them around every minute or so, such that each of the sides gets a chance to brown, but not burn. If they seem to be sticking to the pan, you may want to add more oil.
Once the potatoes are mostly brown, add the cheese. The goal is to have it on the potatoes and not on the skillet. Cover and reduce the heat to low, which is a good tactic in general for melting cheese -- the cheese isn't in contact with the pan so you need something else to heat it up, which is the steam that collects when you've covered the skillet.