Special equipment: Mandoline, 8-inch nonstick saute pan
Preheat the oven to 425 degrees F.
Working 1 potato at a time, slice very thin on the 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 the potatoes. Coat an 8-inch nonstick saute pan with olive oil, using the potato slices starting from the center make concentric circles of potatoes. 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 every 2nd or 3rd layer sprinkle 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 2 potatoes.
Place the pan of potatoes over medium heat and cook until the olive oil begins to sizzle and the potatoes begin to brown on the bottom. Put the pan in the preheated oven and bake for 20 to 25 minutes. Remove the pan from the oven and 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.
Return the cake to the oven and bake for another 10 to 15 minutes or until the cake is fork tender. Cut into wedges and serve.