Allow the reindeer meat to thaw for about half an hour. Fry the slices on a hot pan until the liquid has evaporated. Add some butter to the pan and brown. Season with salt, pepper, and thyme.
Pour the potato and root vegetable strips as well as the sautéed meat into a baking dish, and mix lightly. Pour cream or oat milk into the dish. Sprinkle grated cheese on top of the gratin.
Bake at 200 degrees Celsius for about an hour, until the surface has browned.