Preheat the oven to 375°F (190°C). Butter the bottom and sides of a 9-by-13-inch (23-by-33-centimeter) or a 3-quart (3-liter) baking dish with butter.
In a small saucepan, combine the cream, broth, garlic, sage, and a good pinch of salt and pepper over medium-low heat and heat until small bubbles begin to appear around the edge of the pan and the liquid is steaming, 10 to 15 minutes. Remove from the heat and let steep while you prepare the potatoes.
Cut the rounded ends off each potato, then peel the potatoes. Using a food processor or mandoline fitted with the slicing blade, cut the potatoes crosswise into slices 1/4 inch (6 millimeters) thick. Layer the potato slices, alternating types, in the prepared baking dish. Pour the warm cream mixture through a fine-mesh sieve evenly over the potatoes and discard the garlic and sage. Cover the dish tightly with foil and bake for 45 minutes.
In a small bowl, toss together the Parmesan and Gruyère cheeses.
Uncover the potatoes and sprinkle with the cheeses. Continue to bake, uncovered, until the mixture is bubbly and thick, 25 to 30 minutes. Let stand for 10 minutes before serving.