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Quarter the potatoes lengthwise and thinly slice them crosswise. In a large skillet, heat all but 2 tablespoons of the olive oil until shimmering. Carefully add the potatoes to the skillet and cook over high heat, stirring once or twice, until the potatoes are softened and just beginning to brown, about 15 minutes. Using a slotted spoon, transfer the potatoes to a colander to drain. Pour off the oil in the skillet.
Return the skillet to high heat and add the remaining 2 tablespoons of oil. In a large bowl, combine the potatoes with the beaten eggs and season with salt. Add the mixture to the skillet and cook, lifting the edge of the tortilla and tilting the pan to allow the uncooked eggs to seep underneath; cook until the bottom is golden and set, about 5 minutes. Slide the tortilla onto a large plate and pour off any oil. Cover the tortilla with the inverted skillet; using 2 oven mitts, grasp the skillet and the plate and quickly invert the tortilla back into the pan. Cook the tortilla over high heat until the bottom is set and lightly browned, about 2 minutes longer. Slide the tortilla onto a plate, cut into wedges and serve.