Special equipment: a deep-frying thermometer and a 3-ounce travel spray bottle
Put the oil in a medium pot and heat over medium-high heat until a deep-frying thermometer registers 325 degrees F. Line a plate with paper towels and put it next to the pot. Add the vinegar to a 3-ounce travel spray bottle and set aside.
Line another plate with paper towels, put the carrots on the plate and pat with additional paper towels to dry thoroughly. This will prevent the oil from splattering when the carrots are added.
Working in batches, fry the carrots until light brown and crispy, 2 to 3 minutes. Transfer to the paper towel-lined plate.
Season the carrot fries with salt, sprinkle over the mint and orange zest and spray with a couple of spritzes of cider vinegar.