Using a French fry cutter, cut the ends off each end of the potato to fit inside the cutter. Pressing down on the handle, push the potatoes through the cutter blade. (If you do not have a French fry cutter, use a knife to cut the potatoes into sticks 3/8" x 3/8" x 3" long.)
Rinse the cut potatoes in a large bowl under cold running water until the water turns clear. Cover with ice water and refrigerate at least 30 minutes and up to 1 day in advance.
Heat the peanut oil in an electric deep fryer to 325 F according to manufacturer's instructions. Line a baking sheet with a brown paper grocery bag. Have another brown bag ready for later use.
Pour off the ice water and thoroughly dry the potatoes in a clean kitchen towel. Place the potatoes into the fryer basket and immerse in the hot oil. Fry the potatoes until they are softened throughout and a light golden brown, but not too crispy, 5 to 7 minutes. Remove the basket from the oil and let any excess oil drain back into the fryer. Transfer the potatoes to the prepared baking sheet.
Increase the heat on the electric deep fryer to 375 F.
Place the potatoes back into the fryer basket and immerse into the heated oil. Discard the first brown bag and replace with the second brown paper bag. Fry potatoes, shaking the fryer basket occasionally, until the potatoes are golden brown and crispy, about 1 to 2 minutes. Transfer the fries from the hot oil to the brown paper bag and let drain. Immediately season the fries to taste with salt.