Using a mandoline or a very sharp knife, slice the potatoes into rounds as thin as possible, and place in a large bowl of water to prevent discoloration. Heat the oil in a large, heavy pot to between 340 degrees F to 350 degrees F. Pat the potatoes completely dry and fry in batches, cooking until golden brown, stirring with a long handled spoon to turn and cook evenly, about 2 minutes. Drain on paper towels and place in a large bowl. Sprinkle with the salt and pepper and serve with the Cilantro Aioli.
Place the egg, Dijon, garlic, and cilantro in a food processor, cover, and process until smooth. Slowly drizzle in 1/2 cup of the olive oil through the feed tube in the top of the blender in a slow but steady stream, until mixture begins to emulsify. Add the lime juice and continue drizzling in the remaining olive oil. Turn the blender off, add salt and pepper, and pulse to incorporate. Chill if not using immediately.