Preheat oven to 400F. Butter a 1 1/2-quart shallow baking dish.
Bring a large saucepan of salted water to a boil.
Peel potatoes and cut into 1/8-inch slices.
Add potatoes to boiling water and cook 4 minutes.
Drain potatoes thoroughly in a colander.
Arrange 1/3 of the potatoes in baking dish, top with 1/3 of the cheese, 1/3 of the thyme leaves, and sprinkle with salt and freshly ground pepper. Repeat twice more, ending with a layer of cheese and thyme.
Heat milk just to boiling.
In a small bowl whisk egg and add hot milk, stirring. Pour evenly over potatoes.
Bake 30 minutes, until top is golden and potatoes are tender.