How to Make It
Preheat the oven to 350° F and pull out a large cast iron pan; I use an 8-inch pan.
Thinly slice the potatoes and toss them with the pimentón, cayenne, and salt. Meanwhile, preheat the cast-iron pan over medium-high heat.
When the pan is very hot, add the oil; it should immediately begin to shimmer. Carefully add a slice of potato to the hot oil. If the oil begins to bubble vigorously around it, add the rest of the potatoes; if not, wait a few more minutes for the oil to really heat up.
Fry the potatoes in the oil until they are just tender, turning them every few minutes so that they cook evenly. While they fry, crack the eggs and whisk them together in a medium bowl.
Remove the fried potatoes to a bowl and drain all but 2 tablespoons of the oil very! carefully! into a glass jar or an aluminum can. (You can reuse this oil to fry other things, if you like, as long as you don’t mind that they’ll taste like pimentón—chickpeas or carrots or both would be very good!)
Return the pan to the stove over medium heat, add the tomato paste, and stir well to combine until the oil and the tomato paste are more or less unified. Pour the tomatoey oil over the potatoes, toss gently to combine, and then add the potatoes back to the pan. Pour the eggs over the potatoes, then set the pan in the oven.
Bake for 20 to 25 minutes, until the center is just set. Let cool until warm, lukewarm, or cold, then invert the whole shebang onto a large plate or serving board. (Gently loosen the tortilla from the sides and bottom of the pan with a knife or a spatula first.) Slice into wedges or squares and serve.
If you’d like, serve it with a dollop of garlicky mayonnaise: Sprinkle the minced garlic with a big pinch of salt and use the edge of your knife to scrape it into a paste by pressing the edge of the blade down and pulling it across your cutting board.
Stir the garlic into the mayonnaise and let it sit for at least 10 minutes before serving with the tortilla.