My wife prefers this approach to Keller's recipe from Bouchon. I, however, have to believe that if Thomas Keller himself made his recipe he would outdo this creation of mine by a wide margin.
Peel the potatoes and place in a bowl of cold water.
Using a mandoline, slice the potatoes to a uniform thickness. They should be about 2-3 mm thick. Place the slices in a bowl of cold water and set aside.
Melt the butter in a medium saucepan over medium heat and as the foam subsides add the flour and stir together to make a roux.
Stir a few tablespoons of milk into the roux to thin it. Gradually add the rest of the milk and stir to combine into a smooth sauce.
Season with salt and pepper to taste. It is best to overseason just a bit at this stage, since the the sauce will be combined with the unseasoned potatoes later.
Mix the cheeses together and one handful at a time, add the cheese to the sauce and stir to melt and combine. Continue until all but a cup of cheese has been used. Set that aside.
Drain the potatoes in a colander.
Drizzle the sauce into the bottom of a deep casserole. There is no need to create a thick uniform layer, as the sauce will spread out during cooking.
Cover the sauce with a layer of overlapping potato slices.
Drizzle more sauce over the potatoes. You do not need a full layer of sauce to completely cover the potatoes. If you do, your final dish will be too liquid. You really want just enough sauce so that the weight of the potatoe above will cause it to spread out into a very thin layer. The potatoes should end up bound together by the sauce, not swimming in it. Repeat the process until the potatoes are all used up. Depending on the shape of your backing dish, you should get 5-10 layers of potatoes.
Drizzle a final layer of sauce over the last of the potatoes, then sprinkle the top with the remaining cheese. Don't worry if you have extra sauce. You can refrigerate it and reheat it to use on vegetables or macaroni and cheese.
Place in a preheated 350 degree F oven. Bake approximately one hour or until the top is golden brown and bubbly around the edges and a toothpick or skewer inserted in the center easily penetrates the potatoes and comes out clean.