Special equipment: a mandoline
Preheat the oven to 425 degrees F.
Working 1 potato at a time, slice the potatoes very thin on a mandoline. (Work quickly to prevent discoloring, and don't put the potatoes in water, as this will wash off the starch.) Coat an 8-inch nonstick saute pan with olive oil, then layer the potato slices, starting in the center of the pan and making concentric circles. (Remember that eventually the bottom will become the top, so it is important for the first circles of potato to look pretty.) Brush each potato layer with olive oil, and sprinkle every second or third layer liberally with grated Parmigiano and salt. After each layer, press the potatoes so they are very compact.
Place the pan over medium heat, and cook until the olive oil begins to sizzle and the potatoes begin to brown on the bottom. Transfer the pan to the oven and bake about 25 minutes. Remove the pan from the oven, and cover with a tight-fitting lid. Carefully and completely drain the excess oil out of the side of the pan (drips may cause burns). Flip the pan so the potato cake is turned out onto the bottom of the lid, then slide the potato cake off the lid and back into the pan (so the brown side is now facing up and the top becomes the bottom).
Return the cake to the oven and bake until fork-tender, another 10 to 15 minutes. Cut into wedges and serve.