How to Make It
In a very large saucepan, cover the potatoes with water. Generously season with kosher salt and bring to a boil. Simmer the potatoes until nearly tender, about 10 minutes. Drain and let the potatoes dry on paper towels. Transfer to a work surface and halve each potato lengthwise.
Meanwhile, in a medium bowl, whisk the egg yolks with the garlic, lemon juice and zest. Gradually add the 1 cup of oil in a very thin stream, whisking constantly, until a thick, creamy sauce forms. Whisk in the paprika and season the aioli with kosher salt.
In a large saucepan, heat 2 inches of vegetable oil until it reaches 350° on a deep-fry thermometer and line a baking sheet with paper towels. Working in batches, fry the potatoes until completely tender and lightly browned in spots, about 5 minutes. Using a slotted spoon, transfer the potatoes to paper towels to drain. Sprinkle the potatoes with parsley and Maldon salt and transfer to a bowl. Serve right away with the smoky aioli.