Preheat oven to 425°; butter a shallow baking dish or gratin.
Bring the sliced potatoes, milk, wine, salt, and pepper to a boil in a large pan, stirring occasionally; leave at a simmer while you get on with the mushrooms.
Melt the butter and garlic oil in a skillet over med-high heat; add in mushrooms and cook, stirring occasionally, until softened, about 3 minutes.
Pour the mushrooms and their garlicky juices into the pan of potatoes, stir to mix.
Transfer to the gratin dish; bake for 45 minutes or until piping hot, and crisp on top.