Using a small slicer or a mandoline, slice the garlic very thin.
Heat the olive oil in a 10-inch cast-iron skillet over medium-high heat until very hot. Line a plate with two paper towels. To test the temperature of the oil, add a slice of garlic. If it sizzles, add the rest of the garlic and cook until just crisp and light golden brown, a matter of seconds. Use a flat slotted skimmer to keep the slices from sticking together as they cook, and transfer them to the paper towels to drain the moment they turn color. (The oil can be strained and used for another batch or reserved for another use.)