Peel and rinse the potatoes. Cut each potato lengthwise into 4 or 5 pieces, then cut each piece into sticks. The thinner these are, the crispier they will be. Place the fries in a large bowl. Cover with cold water, then allow them to soak 2 or 3 hours (or you can stick them in the fridge and let them soak overnight).
When you're ready to make the fries, drain the water and lay the potatoes on 2 baking sheets lined with paper towels. Blot with paper towels to dry.
Heat a few inches of vegetable oil to 300 degrees F in a heavy pot. In 3 or 4 batches, fry the potatoes about 4 to 5 minutes per batch, or until soft. They should not be brown at all at this point-you just want to start the cooking process. Remove each batch and drain them on new, dry paper towels.
Once all the potatoes have been fried at 300 degrees F, turn up the heat until the oil temperature reaches 400 degrees F. When the oil is hot, start frying the potatoes in batches again, cooking until golden and crisp. Remove from the oil and drain on fresh paper towels. Sprinkle the fries with sea salt and dive in with the ketchup-mayo mixture.