One way of frying potatoes is to parboil them before crisping them in shallow fat. It's a fiddly process, and the potatoes break up easily -- all to the good, some might say. An easier method is to treat them as you would chips: submerging them in fat, cooking them gently until they are soft, and turning up the heat to brown them. It works deliciously with both new and maincrop potatoes.
Even the largest, 28 cm frying pan will accommodate only about four medium potatoes in a single layer. You need the best part of half a litre of oil -- I use sunflower. Pour it into the pan, and set the pan on a low to medium heat. Peel the potatoes, and cut them into 1 cm cubes. Wipe them with a paper towel to absorb some of the moisture, which would cause the oil to bubble alarmingly; but, provided the oil is not too hot, a little moisture will not cause an overflow. Drop a small piece of bread into the oil; when it sizzles, you can tip in the potatoes. Once they are cooking, you may find that you can turn down the flame further.
The potatoes should be soft in about 15 minutes. Turn up the flame to medium/high. The potatoes will brown in about five minutes. You may need to turn some of them.
Lift them out of the oil with a slotted spoon, drain them on kitchen paper, and sprinkle them with salt.