Peel potatoes and place them, whole, in a bowl of cool water to prevent browning. Use a sharp knife or a crinkle cut knife to cut the potatoes into ½- by ½- by 2 ½-inch batons.
Fill a medium pot of water, add the kosher salt and the potatoes, and bring to a gentle simmer. Cook potatoes until you can pierce one with a knife with little resistance, 10–12 minutes. Use a slotted spoon or spider strainer to carefully transfer them to a plate to cool to room temperature.
Pour oil into a large Dutch oven or deep-fat fryer and heat oil to 250°. Fry the potatoes, in two batches, until they are slightly colored and have developed a thin crust, 5–6 minutes. Use a slotted spoon or spider strainer to transfer them from the oil and to a paper-lined plate to drain.
Heat the oil to 370° and fry the potatoes until they are golden brown and crispy, 5–6 minutes. Use a slotted spoon or spider strainer to transfer them from the oil and to a paper-lined plate to drain. Season with sea salt and pepper and serve with aioli.