Fill a bowl with cold water. Slice the potatoes lengthwise into 1/4-inch-thick slices, then stack the slices and cut into 1/4-inch-wide sticks. Transfer the potato sticks to the water as you go. Drain the potatoes and rinse under cold running water until the water runs clear. Spin in a salad spinner to remove the excess water, then pat dry.
Heat 3 to 4 inches peanut oil in a wide, deep pot over medium-high heat until a deep-fry thermometer registers 300 degrees F. Fry the potatoes in 2 batches until about halfway cooked, 3 to 4 minutes per batch. Remove with a slotted spoon and transfer to paper towel-lined baking sheets to drain.
Increase the oil temperature to 365 degrees F. Fry the potatoes again in 3 or 4 batches, turning and stirring with tongs, until golden brown, about 3 minutes per batch. Transfer to fresh paper towels to drain and season with salt. Sprinkle with Old Bay and vinegar.