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Get out a 9 x 12-inch (22 x 30-cm) baking dish and rub 1 tablespoon of butter over the bottom and insides to prevent the potatoes from sticking. Set the dish aside while you prepare the potatoes. Once you've sliced all of the potatoes and put them in the bowl, slowly pour the warm cream mixture over the slices. Use your hands or a spoon to toss the potatoes in the mixture so they're completely coated. Use the spoon to push the potato slices down. This way the cream should cover most of the potato slices. Scatter 3 ounces (85 g) of shredded Gruyère cheese evenly over the top of the potatoes. Place the baking dish in the center rack of the preheated oven. Set a baking sheet on the rack that's underneath it to catch any drips. Bake the potatoes until they're soft when you insert and remove a fork. The cream mixture should bubble and the cheese will brown once it's finished cooking. If the potatoes are still firm, cook them for 15 minutes and check them again. Turn off the oven and let the au gratin potatoes sit for 5 to 10 minutes before you serve them. This will make it easier to handle them and the liquid will thicken a little as it cools. Serve the au gratin potatoes with sliced ham, greens, or roasted vegetables. Store the leftover au gratin potatoes in an airtight container in the refrigerator for 3 to 5 days.
Preheat the oven to 375 °F (191 °C) and grease a baking dish. Mix the potato slices with the cream mixture. Transfer the potato mixture to the dish and sprinkle the cheese on top. Bake the au gratin potatoes for 1 hour and 15 minutes. Serve the au gratin potatoes.