Fill a large pot with about 1 1/2 inches of equal parts olive oil and vegetable oil. Line a baking sheet with paper towels.
Slice the potatoes on the thinnest setting of a mandoline (they should be almost see-through). Lay the slices on a clean kitchen towel in a single layer and pat dry with paper towels. One at a time, transfer the slices to the cold oil (starting with cold oil keeps the potatoes from sticking to one another). If the potatoes are not completely submerged, add more oil to cover. Add the garlic and rosemary to the pot.
Turn the heat to medium high and cook, undisturbed, 5 minutes. Stir the potatoes once to make sure they are not sticking to the bottom, then let fry, undisturbed, until golden, 18 to 20 minutes. Remove the chips with a strainer and drain on paper towels. Season with salt.