Step 1Cut the potatoes into 1/8-inch-thick slices and pat dry.
Step 2Heat about 1/2 inch of olive oil in a large, preferably cast-iron, skillet over high heat. The oil should be hot but not smoking. Test it by dropping in one potato slice; it should "boil," as the Greeks say, throwing off lots of bubbling foam. Add just a handful of potatoes at a time, no more than can fit in one layer in the skillet.
Step 3Cover the skillet and reduce the heat to medium. This is a particularly Greek home-cooking technique. The potatoes fry and "boil" in the oil, thus cooking on the inside thoroughly.
Step 4After about 6 or 7 minutes, carefully lift the lid, and with a two-pronged long fork check the bottom of the potatoes. They should have begun to acquire some color. Gently turn the potato slices, one by one if possible, to fry on the other side. Fry until deep golden-yellow and just lightly browned. Remove with a slotted spoon, drain in a colander or on paper towels and continue, adding more olive oil to the skillet if necessary, until all the potatoes are fried.
Step 5To serve, place the potatoes on a large platter and toss with salt to taste, the oregano and grated cheese, if using. If you like your potatoes crunchy, do not squeeze lemon over them; if you don't mind a soft fry and like the tang of lemon, then douse them according to taste.