In a large heavy-bottomed saucepan, pour enough oil to fill the pan about a third of the way. Heat over medium heat until a deep-frying thermometer inserted in the oil reaches 365 degrees F. Line a sheet tray with paper towels.
Cut the potatoes using the thin julienne attachment of a mandolin. Rinse the potatoes in water, then spin dry in a salad spinner. Make sure the potatoes are dry so they don't spatter in the hot oil. Fry in batches until crisp, 4 to 5 minutes. Remove the cooked fries with a slotted spoon and drain on the prepared sheet tray. Season with salt immediately. Make sure the oil heats back to 365 degrees F between batches of fries. Repeat for the remaining potatoes.
In a bowl, mix the mayonnaise, garlic and lemon juice. Season with salt and pepper. Serve with the fries.