Preheat the oven to 450° F. Butter a 8x8-inch baking dish.
For the mashed potato, add quartered russet potatoes to a pot with 1½ tablespoons salt and enough water to cover by 2 inches. Bring to a boil over high heat; reduce heat to medium and simmer until tender when pierced with a sharp knife, but not falling apart, 15 to 20 minutes. Drain very well.
Return potatoes to the pot, and mash until almost smooth. Add butter, salt, and cayenne. Mash until smooth.
Add milk and the 4 ounces shredded cheese, and mash and mix until evenly combined. Spread evenly into the prepared baking dish, and smooth the top. Season the top with salt and freshly ground black pepper.
For the au gratin potato layer, slice each half potato into 1/8-inch slices.
Fan the potato out slightly at an angle, and place on top of mashed potatoes. Repeat with remaining potato halves to cover the surface, creating 4 rows across by 2 potato halves long, using 8 halves total. Depending on the size of potato and dish, you may have to adjust the layout to cover.
Generously salt sliced potatoes, and brush generously with melted butter. Scatter the 2 ounces shredded cheese on top, and finish with grated Parmigiano-Reggiano.
Bake in the preheated oven until sliced potatoes on top are tender when pierced with a knife, about 1 hour.