Preheat the oven to 350°F.

Peel the potatoes, then thinly slice them about ⅛-inch thick. Peel, halve, and thinly slice the onion. Shingle the potato and onion slices in layers in a 2-quart to 3-quart casserole dish, seasoning the layers lightly with salt and pepper as you go. ( I used about ½ teaspoon of each when assembling the gratin.)

In a medium pot, melt the butter over medium heat. Whisk in the flour. Cook whisking constantly, for 2 minutes. Drizzle in the warm milk while whisking. Continue whisking until the mixture starts to bubble and thicken slightly, 2 to 3 minutes.

Remove from heat and stir in the shredded cheeses, salt, and pepper until the cheese is melted. Pour the cheese sauce over the potatoes. Cover the pan tightly with foil.

Bake for 1 hour or until a fork can be inserted into the potatoes with slight resistance. (They should feel a little bit firm but not hard. If so, continue baking, checking every 10 to 15 minutes.)

Carefully remove the foil and continue baking for an additional 20 to 30 minutes, or until the top is golden brown and the potatoes are tender. (You should be able to easily insert a fork into the center.) Let the gratin cool for 10 to 15 minutes, so the sauce thickens up slightly, before serving.