Cover the potatoes with water and bring to a boil in a medium pot over medium-high heat. Boil for 4 minutes until just barely tender, and then drain in a colander.
Heat a heavy frying pan over high heat (cast iron is excellent). Once very hot, add enough olive oil to come 1/2" up the sides of the pan and heat the oil to 350°F.
Add the thoroughly drained potatoes to the pan and fry for approximately 15 minutes until golden brown, flipping half way through to brown both sides. Drain and salt immediately. Serve with or slathered in aioli.
In the bowl of a food processor or blender jar, add the egg yolks, salt, garlic, lemon juice, and pepper. Process until homogeneous.
VERY SLOWLY, drop by drop, while the machine is running, drizzle in the olive oil. Extreme patience must be taken with this slow process initially, as too much olive oil poured in at once will cause the sauce to break. After half of the oil has been successfully emulsified, the rest of the oil can be added in slightly faster - but use caution here as too much at once will spell disaster!
Once the olive oil is completely poured in and emulsified, remove the aioli to a sealed container and refrigerate. This condiment is fabulous on coleslaw, sandwiches, egg salad.
** This recipe contains raw egg yolks. If you have a compromised immune system, you can use pasteurized eggs for this recipe.
Makes approximately 2 cups.