Preheat the oven to 450 degrees F and line a baking sheet with foil.
In a small saucepot, bring the butter, oil and garlic to a boil over high heat; remove from the heat.
Using 2 wooden spoons placed on each side of a potato as a guide, cut the potato into thin slices about 1/8 to 1/4 inch thick, without cutting through the bottom (the spoons will keep you from cutting all the way through the potato). Repeat with the remaining potatoes. Place the potatoes on the prepared baking sheet and brush each potato with some of the butter mixture, trying to get the butter between the slices. Sprinkle each potato with salt and pepper.
Bake for 30 minutes, then remove the potatoes from the oven. The potatoes will have started to spread out from cooking and it will be easier to brush each slice with some more of the butter mixture. Put the potatoes back in the oven for an additional 20 minutes.
Remove the potatoes from the oven again and sprinkle the Parmesan over the top and into the slices. Return to the oven to melt and brown the cheese, an additional 5 minutes. Serve the potatoes immediately.