Heat oven to 375 degrees F. Coat a 1-1/2-quart oval (or round) baking pan with nonstick cooking spray.

Place a handheld mandoline ($20, available at oxo.com or at kitchen accessory stores) over a large bowl. Place one potato on the slicer, and secure with guard. Using back-and-forth motion, carefully slice potato into 1/8-inch-thick slices. Alternatively, thinly slice potatoes with knife on a cutting board. Repeat with all potatoes.

Sprinkle flour and salt over potato slices, and toss with your hands to coat potatoes.

Begin layering: Spread 1/3 of the potato slices over bottom of prepared pan, then sprinkle with half the scallions. Top with 1/3 of the shredded Gruyere cheese, about 1/2 cup. Repeat with more potato slices, the remaining scallions and 1/2 cup shredded cheese. Top with remaining potato slices.

In a small saucepan, combine the half-and-half and the nutmeg. Bring to a simmer and remove from heat. Carefully pour over potatoes, then cover pan with foil. Bake at 375 degrees F for 30 minutes. Carefully remove foil, sprinkle with remaining 1/2 cup shredded cheese and the Asiago or Parmesan. Return to oven and bake for 25 minutes, until top is lightly browned and potatoes are tender. Let stand 15 minutes before serving.