Cut the potatoes into irregular 1/8-inch thick slices, so that they take on a homey look.

Heat the oil in a 10 ½-inch nonstick skillet over medium heat until it just starts to shimmer in the pan. To test the heat of the oil, drop in a potato slice; if bubbles form around it, it’s ready. It should not be hot enough to crisp the potatoes.

Stir in the potato slices, coating them well with the oil. Partially cover the skillet and cook for 5 minutes, stirring once and pressing down slightly on the potatoes with the back of a wooden spoon, so that the potatoes begin to color and soften slightly.

After 5 minutes, add the onions, stir well to combine with the potatoes, and continue to cook, partially covered, until the onions wilt, 15 minutes more. Stir every 5 minutes from underneath and press down on the mixture with the back of a wooden spoon to break it up slightly.

Remove the cover and cook 15 minutes longer, stirring well. The potatoes and onions should be very soft and lightly golden but not crusty. During cooking, adjust the heat if necessary to keep it from cooking too fast.

Over a bowl, drain the vegetables in a coarse strainer, tapping the strainer a bit to remove as much oil as possible. Reserve the oil for frying other vegetables.

In a large bowl, lightly beat the eggs. Add the drained potatoes and onions. Season well with salt and pepper. Add about 1 teaspoon of the reserved oil to the skillet and heat over medium heat. Add the egg mixture, reduce the heat to medium-low, and if it begins to stick to the bottom of the pan, shake the pan slightly to move the mixture around or loosen it a bit with a spoon or spatula. Cook until the bottom is pale golden, 4 to 5 minutes, and invert it onto a flat plate.

Add another teaspoon of oil to the skillet if necessary and slip the omelet back in, uncooked side down. Cook until the eggs are cooked but still a bit soft on the inside, another 3 to 5 minutes, depending on the desired consistency.

Invert the omelet onto a serving platter, and if you can beat it, let rest for 5 minutes before cutting into serving wedges and eating.

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