Peel the potatoes and rinse them under cold water. Slice them thinly (about 3-4 mm thick) using a sharp knife or mandoline.
Pat the potato slices dry with paper towels and put them into a large bowl, then sprinkle with salt and mix well.
Heat a ½ inch to an inch of high quality extra virgin olive oil in a large, preferably high-sided frying pan over medium-high heat. When the oil is hot, lower to medium-low. Add the potatoes and add more oil if necessary until all are covered.
Cook the potatoes on medium-low heat for approximately 20 minutes. They may break down a little; this is okay.
While the potatoes are cooking, beat the eggs in a large bowl and season with some salt.
Fry onion slices in oil over medium heat in a separate pan until it begins to caramelize, about 10-20 minutes. Add a splash of water now and then and you can put the lid on to help the process along. Drain off any excess oil allow to cool slightly. Then and add to the egg mixture.
Once the potatoes have fried for 20 minutes, remove them with a large slotted spoon/spider into a colander and allow them to cool and drain for at least 5 minutes.Strain the oil from the potatoes and save it for another use. Add the potatoes to the egg mixture and stir well. Let the egg mixture sit for about 15-20 minutes.
Place your non-stick pan over medium-low heat and pour in the egg mixture.
Cook over medium-low heat for about 6-8 minutes. Make sure the heat is low enough or the tortilla will cook too quickly and be rubbery. You can run a rubber spatula along the edges to make sure the tortilla isn't sticking.
To flip the tortilla, take a large plate and put it over the pan and flip quickly. Some uncooked egg mixture will likely slip out, but that's okay. Slide the tortilla back into the pan to cook the other side, and cook for another 6-8 minutes.
Slide the tortilla out of the pan onto a serving plate and let cool a little before slicing.
Enjoy warm, at room temperature or even cold!