Start by coarsely grating the peeled potatoes and the onion into a clean tea towel. Twist the towel tightly to remove as much liquid as possible from the mixture. Transfer the drained potato and onion mixture to a large mixing bowl.
To the bowl, add the beaten egg along with a generous amount of salt and freshly ground black pepper. It's important to season the mixture well to avoid blandness. Stir thoroughly to combine all the ingredients.
In a heavy-based frying pan, heat a generous amount of vegetable oil over medium-high heat. Once the oil is hot but not smoking, drop spoonfuls of the potato mixture into the pan, flattening each into patties approximately 1cm (½ inch) thick. Fry the patties until they are golden brown and crispy, about 2–3 minutes on each side.
Serve the hash browns hot, as a delightful accompaniment to breakfast or supper. They pair exceptionally well with bacon and eggs.