Peel the potatoes, then cut them in half lengthwise and place flat on the cutting board. Cut into thin half-moon slices crosswise (about ¼-inch thick). Alternatively, you can hold the potato over a bowl and cut off thin, slightly irregular slices with a paring knife, as my mother-in-law does.
Rinse the potatoes in several changes of cold water until the water is clear when you swish them around. Drain and pat dry.
Pour about 1½ inches of oil into a medium skillet and heat it over high heat until shimmering. If you drop a piece of potato into it, it should sizzle upon contact. Add the potatoes, a few pinches of salt, and a little more oil if needed to cover. Stir to coat the potatoes, then lower the heat to medium. Cook until the potatoes are completely tender, all the way to the core (15-20 minutes)—you’re looking to poach them rather than fry them, although a little browning around the edges once the potatoes are fully cooked won’t hurt (depending on who you ask). Rotate them delicately from time to time as they cook, and don't worry if they begin to break apart a bit (you just don't want an oily puree). If you're adding onions, add when the potatoes are partially cooked, about halfway through the cooking time.
Using a slotted spatula or spoon, transfer the potatoes to a colander set over a bowl to let them drain. Taste for salt and sprinkle with a bit more if you wish. When the oil has cooled, strain it through a fine-mesh sieve and save it in a jar to make your next tortilla. Wipe the skillet clean.
Place the eggs in a bowl and add one pinch of salt per egg. Whisk until well blended. Add the potatoes and gently stir until coated.
Heat 1 tablespoon of the potato-poaching oil in the skillet over medium-low heat and pour in the potato and egg mixture. Using a flexible spatula, tuck the egg in around the edges of the skillet to make a rounded side and to ensure the tortilla isn’t sticking. As the tortilla cooks, shake the pan a bit to prevent sticking. Cook until the egg is set and lightly golden on the bottom but still a bit runny on top (about 5 minutes).
Okay, here it comes. Invert your plate of choice over the skillet and place your hand firmly over the top. Grab the skillet handle in the other hand, lift it off the burner, and flip it over quickly and decisively. If all has gone well, your tortilla is now on the plate. Return the skillet to the burner and slide the tortilla back in with the uncooked side down. Tuck in the edges again and continue to cook for 4 to 5 more minutes, until the tortilla is just set in the center, or done to your liking.
Either flip or slide the tortilla out onto a clean, dry plate.
Serve warm or at room temperature, cut into wedges or mini squares.