Gruyere Potato Gratin

I love cheese, so if I'm given an opportunity to add it to my food, I'm all over it.

1 pound large red potatoes
3/4 cup coarsely grated Gruyere (about 3 ounces)
1 large egg
3/4 cup milk, heated just to boiling
Salt and freshly ground black pepper

Preheat oven to 400 degrees F. Butter a 1-quart gratin or shallow baking dish.
Bring a large saucepan of salted water to a boil. Peel potatoes and cut into thin slices (about 1/8-inch thick). Add potatoes to boiling water and cook approx. 4 minutes (a bit longer in high altitude). Drain potatoes well in a colander.
In gratin or baking dish, arrange potatoes, overlapping them, in 3 layers, sprinkling first 2 layers each with 1/4 cup Gruyere and salt and pepper, to taste.
In a small bowl whisk egg and add hot milk in a stream, stirring constantly. Season mixture with salt and pepper and pour evenly over potatoes. Sprinkle the potatoes with remaining 1/4 cup cheese and bake until top is golden and potatoes are tender, about 30 minutes