Preheat the oven to 400 degrees F. Butter a large baking dish with the butter.
Slice the potatoes into sticks, and then cut the sticks to create a dice.
Combine the cream and milk in a bowl. Add the flour, salt and some pepper. Whisk it together well so that the flour is incorporated into the milk/cream mixture. Add the diced potatoes to the prepared baking dish and pour the creamy mixture all over the top.
Cover the dish with foil and bake for 20 to 30 minutes. Remove the foil and bake 15 to 20 minutes more. Just before serving, sprinkle on the grated cheese and return it to the oven until the cheese is melted and bubbly, 3 to 5 minutes. Sprinkle on the green onions and serve it hot. Creamy, dreamy and divine.