In a small bowl stir together the butter and the oil. In a food processor fitted with a 1-millimeter slicing blade or with a mandoline or similar hand-held slicing device, slice the potatoes thin. Working quickly to prevent the potatoes from discoloring, brush the bottom of a 9-inch cast-iron skillet with some of the butter mixture and cover it with a layer of the potato slices, overlapping them. Brush the potatoes with some of the remaining butter mixture, sprinkle them with some of the rosemary, and season them with salt and pepper. Layer the remaining potatoes with the remaining butter mixture and the rosemary in the same manner.
Heat the mixture over moderately high heat until it begins to sizzle, transfer the skillet to the middle of a preheated 450°F oven, and bake the galette for 25 minutes, or until it is golden and the potatoes are tender. Cut the galette into wedges.