Have ready a large bowl of cold water and a deep-fry/candy thermometer. Line a large platter or baking sheet with several layers of paper towels. Peel the potatoes, cut into 1⁄4-inch-thick fries, and place in the bowl of cold water.

Place the peanut oil in a 6- to 8-quart stockpot and heat it to 300 degrees over medium-high heat. Pat very dry with paper towels as many potatoes as will fit comfortably in the pot, add them to the pot, and cook until they just begin to brown, about 5 minutes. Transfer to the paper towels to drain, and repeat with the remaining potatoes, frying them in batches. Let stand for 30 minutes. (Set the oil aside.)

To make the sauce

Stir together the mayonnaise and ketchup, and set aside.

Line another baking sheet with several layers of paper towels. Reheat the oil to 350 degrees (175°C). Add the potatoes, again in batches, and cook until golden brown, about 3 minutes. Do not crowd them in the pot, and gently separate them with a wooden spoon as they cook so they don’t stick together. Transfer the cooked potatoes to the paper-towel-lined baking sheet and sprinkle with salt. Serve with the sauce on the side.