Grease the bottom and sides of a 1 1/2-quart gratin dish with the butter.
Make a layer of overlapping potato slices on the bottom of the dish and sprinkle with half of the garlic and 1/3 of the goat cheese. Season with salt and pepper. Repeat with another layer of potatoes, the remaining garlic, another 1/3 of the cheese, and season with salt and pepper. Finish with a final layer of potatoes and the remaining cheese and season with salt and pepper. Pour the heavy cream over the top of the potatoes and, using your hands, press to condense the layers slightly. Make sure the potatoes are all covered with cream.
Bake for 1 hour and 20 minutes, or until the gratin is bubbly and golden brown on top and the potatoes are cooked through. Let cool for 5 minutes before serving.