Preheat oven to 350 degrees F.
Melt the butter in a small saucepan.
Heat a medium nonstick or well seasoned cast iron skillet over medium to medium-low heat. With a sharp knife or box grater, slice the potatoes as thinly as possible. Pat the potato slices dry on a kitchen towel. Start to arrange 1 layer of overlapping potato slices over the bottom of the pan in concentric circles. Pour the clear liquid of the butter over the sliced potatoes, leaving the milky solids on the bottom of the saucepan. Repeat 2 more times, making 3 layers. Drizzle each layer with a bit of the butter and season the layers, alternating salt, pepper, and then nutmeg. Cook on low heat until potatoes are golden on the bottom and crisp around the outside, about 25 to 30 minutes. Shake the pan back and forth several times while cooking to keep from sticking.
Pour off any excess butter into a small bowl and reserve. Place a flat pan lid, the diameter of the skillet, over the potatoes. Holding the lid firmly in place, gently flip over. Add the reserved butter to the pan and slide the potatoes back in. Put the skillet in the oven and bake for 25 to 30 minutes. Shake the pan back and forth several times while cooking to keep the potatoes from sticking. The bottom should be browned and crisp and the potatoes cooked through. Pour off any fat remaining in the pan and slide the potato cake onto a serving dish. Slice into wedges, sprinkle some grated Parmesan cheese and serve.