Peel and thinly slice the potatoes. Soak slices in water for 10 minutes. Drain and dry completely in a salad spinner. Meanwhile, pour enough olive oil into a large saucepan so it’s about an inch deep. Heat over medium heat until a slice of onion bubbles gently when placed in the oil, about 310 degrees F. Add the onions and fry, stirring occasionally until just golden, about 8 minutes. Scoop the onions from the oil with a slotted spoon and drain on a paper-towel lined baking sheet.
Add about half of the potatoes, and fry until lightly golden and a little puffy, about 10 minutes. Scoop the potatoes from the oil, add to the onions. Repeat with the rest of the potatoes. Reserve the oil. Season the vegetables with 3 teaspoons of the salt. (The preparation of the vegetables can be done up to 2 hours ahead.)
Break the eggs into a large bowl. Whisk just enough to combine but not so much that they get loose and runny, about 15 times. Add the vegetables and remaining 1 teaspoon salt.
Heat a medium (10-inch) nonstick skillet over medium heat. Add 2 tablespoons of the reserved oil and heat. Add the tortilla mixture and stir briskly with a spatula to get peaks and valleys in the folds of the egg, about 2 to 3 minutes. Stop stirring and reduce the heat to low, cover and cook until the eggs are set but still pretty loose, about 5 minutes. Lay a flat pan lid or a plate on top and invert the tortilla. Add more oil to the pan if needed, and slide the tortilla back into the pan, cover and finish cooking until set and lightly brown, about 3 minutes more. Slide the tortilla out of the pan to a cutting board to cool slightly.
Cut into cubes or thin wedges and serve warm or room temperature as is or with 1 of the suggested toppings.
Cook’s Note: Yes that’s a lot of oil. But you can reuse it for other tortillas and in sauces, dressing or salads. It has a wonderful onion-y flavor that is great tossed on roasted potatoes or chicken.
We love this as a topper to grilled bread, with cheese, or as an alternative to cocktail sauce with shrimp or other seafood.