Peel the potatoes and cut them into small pieces (about an inch). Heat the oil in a sauté pan and add the potatoes. Fry them for 4 minutes, turning them frequently until they are lightly browned.
Season the potatoes with salt and pepper, lower the heat and cover the pan. Let the potatoes cook for 20 minutes, flip halfway through to cook both sides.
Turn up the heat and sprinkle with paprika and cayenne pepper. Stir well so that they are well covered.
Combine the vinegar and tomato puree and pour into the pan. Be careful as it can splash. Cook for two minutes, constantly stirring until the potato pieces are well covered with a thick and spicy sauce. There should be no liquids or loose oil in the pan.
Transfer the potatoes to a serving dish and drizzle with aioli if desired.
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