Arrange half the kadaif in a buttered baking dish and sprinkle it with chopped walnuts. Put the rest on top and cover with pieces of butter. Bake on a low oven until pink.
Boil the milk with the sugar, remove from the heat, flavor with vanilla and pour it over the baked kadaif. Cover the dish. The next day, turn it over onto a suitable container. Serve cut into squares.