To make the allioli: In a small saucepan combine the garlic cloves and olive oil and bring to a simmer over medium heat. Cook for 5 to 7 minutes, or until the garlic is tender and honey gold (occasionally tilt the pan as needed to keep the garlic submerged–even off the heat the olive oil should be hot enough to keep cooking it). Watch the garlic carefully so that it does not overcook. Life the garlic cloves from the oil with a slotted spoon and set aside on paper towels to cool. Reserve the garlic oil for another use (1 1/2 teaspoons will be used later for the allioli).
In a small food processor fitted with the metal blade, puree the garlic cloves, mayonnaise, lemon juice, and 1 1/2 teaspoons of the garlic oil. Taste, season with salt and pepper, and pulse to mix. Scrape the allioli into a lidded storage container and chill for up to 3 days. You should have about 1/2 cup allioli.
To make the bravas sauce: In a large saucepan, heat the extra-virgin olive oil over medium heat. Add the onions and cook for about 10 minutes, or until they soften but have not colored. Add the garlic and cook gently for 6 to 8 minutes, or until the garlic is tender and aromatic. Stir in paprika, the cumin, and the cayenne.
Add the tomatoes, and the vinegar to the saucepan and season to taste with salt and pepper. Bring to a simmer over medium heat and cook for about 20 minutes, covered, or until the sauce is heated through and the tomatoes have begun to break down.
Let the tomato sauce cool a little and then, working in batches if needed, puree in a blender until smooth. As one batch is pureed, transfer it to a bowl or a container with a tight-fitting lid. Adjust the seasoning as needed. Use the tomato sauce right away, refrigerate it for up to 3 days or freeze it (like you would with any other tomato sauce).
To make the patatas: Preheat the oven to 450 degrees F.
Put the potatoes in a large pot and add enough cold water to cover by an inch. Season generously with kosher salt and bring to a boil over high heat. Reduce the heat slightly and simmer for about 20 minutes, or until just fork tender. Drain the potatoes and set aside to cool. The potatoes can be boiled and cooled a day in advance (be sure to refrigerate them if you do this in advance). When cool, cut the potatoes into wedges, cubes, or if using fingerling potatoes, cut them in half lengthwise.
Season the cooled potatoes with salt and pepper and toss them with a generous drizzle of olive oil in a shallow baking pan (you can use some of the leftover garlic-infused olive oil as well). Spread out the potatoes in the pan so they are an even layer, and not overlapping. Roast for 12 to 15 minutes, or until the potatoes are nicely browned on one side. Rotate the pan and turn the potatoes over. Roast for another 8 to 10 minutes, or until browned and crisp on the other side (I actually roasted mine for a total of 30 to 35 minutes).
If you prefer fried potatoes, heat about 10 cups of canola or another vegetable oil in a deep, heavy pot until it registers 375 degrees F on a deep-fry thermometer. Meanwhile preheat the oven to 250 degrees F and line a baking sheet with several layers of paper towels.
Fry the potatoes in batches so that you don’t crowd the pan. Carefully submerge them in the hot oil and let them cook for 4 to 6 minutes, or until crispy and browned. Lift the potatoes from the oil with a slotted spoon and let them drain on the paper towels. Sprinkle with salt and pepper and keep warm in the oven while you fry the remaining potatoes.
Serve the potatoes with the bravas sauce and the allioli drizzled over the top.