I was in Spain last year and I had several of these while I was there. They are a perfect lunch or even dinner.
3 cups plus 1 Tbsp. extra-virgin olive oil, divided
2 lb. Yukon gold potatoes , peeled and cut into 1⁄8-inch slices
5 large eggs
1 small vidalia onion (7 oz.), medium dice
2 tsp. kosher salt
Line a large plate with paper towels and set by the stove. In a large, heavy-bottomed pot set over medium-high heat, heat the oil until it begins to smoke. Add the potatoes and cook, stirring occasionally and adjusting the heat as necessary to maintain a slow, steady simmer. The idea is to softly “boil” the potatoes in the oil, not fry or brown them. When the potato slices are completely soft and beginning to fall apart, 45–60 minutes, use a mesh strainer or spider skimmer to transfer them to paper towel-lined plate. Reserve the oil, off the heat, in the pot and set the potatoes aside to cool until just warm to the touch.
Meanwhile, in a 9-inch nonstick skillet, heat the remaining 1 tablespoons of olive oil over medium-high heat. When the oil is hot, add the onions, cook, stirring constantly for a minute, then lower the heat to medium and cook, stirring occasionally, until soft and golden brown, 18–20 minutes. Transfer the onions to the lined platter with the potatoes and set aside to cool slightly.
In a large bowl, combine the eggs and salt and whisk until slightly frothy. Add the potatoes and onions and fold together with a large spoon or rubber spatula. In the nonstick skillet, heat 1 tablespoon of the reserved potato-cooking oil over high heat until smoking. Add the egg mixture and cook until the edges begin to set and small bubbles start to form on the surface, 1–2 minutes. Place a wide, flat plate (ideally one that fits snugly inside the rim of the pan) over of the omelet, and flip the pan upside down, keeping the plate level to avoid spillage. Use a rubber spatula to carefully slide the omelet back into the pan, uncooked-side down. Return to high heat, cook for a additional minute, then immediately remove from the heat–it will still be quite liquid at the center. Use the spatula to slide the omelet onto a serving plate, then tent with foil for 5 minutes to allow the soft eggs to set up slightly. Serve immediately or at room temperature.