Preheat the oven to 350°F/177°C.
With a mandoline, cut the potatoes into ⅛-inch slices, or slice thinly with a knife. They can be cut ahead of time and held submerged in cold water for up to a day.
Place the cast-iron gratin dish over medium heat. Add the butter and shallot, and cook until the shallot is translucent.
Layer half the potatoes in the dish. Pour 1 cup of the half-and-half over the potatoes. Season generously with salt and pepper, and judiciously with nutmeg—3 or 4 gratings should do.
Cover the potatoes with half of the cheese. Repeat the layers with the remaining ingredients. Cover the dish with foil and bake until the potatoes are tender, 45 to 60 minutes. This step can be done up to 3 days before serving.
To finish the potatoes, remove the foil, turn up the heat to 415°F/213°C, and continue cooking until the top is golden brown, or put them under the broiler until the cheese is bubbly and golden brown.