Working 1 potato at a time, slice very thinly on a mandoline. It is important to work with purpose here (quickly) because the potatoes will discolor. Do not put the potatoes in water because this will wash the starch off of them.
Coat an 8-inch ovenproof nonstick saute pan with olive oil. Starting from the center of the pan, make concentric circles of potato slices. Remember that eventually the bottom will be the top, so it is important for the first circles of potatoes to look pretty. Brush each layer with olive oil and sprinkle every 2nd or 3rd layer liberally with grated Parmigiano and salt. After every layer of potatoes, press the potatoes so they are very compact. Repeat this process with the remaining potatoes.
Place the pan over medium heat and cook until the olive oil begins to sizzle and the potatoes begin to brown on the bottom, about 10 minutes.
Place a tight-fitting lid on the pan. Drain the excess oil out the side of the pan. THIS STEP IS EXTREMELY IMPORTANT!!! After the excess oil has been drained, completely flip the pan so the cake is turned out onto the bottom of the lid, then slide the cake off the lid back into the pan so the brown side of the cake is now up and the top becomes the bottom. We see now why draining the excess oil is important: if you attempt this without draining the oil, it will drain out on your wrist and burn you.
Transfer the pan to the oven and bake until the cake is fork-tender, 20 to 25 minutes. Cut into wedges and serve.