Using a food processor with a slicing disk, cut the potatoes 1/8 inch thick. Transfer to a large bowl of cold water. Slice the sweet potato 1/8 thick and transfer to a plate.
In a small skillet, melt the butter over medium heat. Add the onion and white scallion parts and cook, stirring occasionally, until tender, about 5 minutes. Stir in the flour and cook for 1 minute. Gradually whisk in the cream and bring to a simmer. Whisk in 1 1/4 teaspoons salt and 1/2 teaspoon pepper and simmer, stirring often, for 1 minute. Stir in the green scallion parts, cover and remove from the heat.
Grease the bottom and sides of a small Dutch oven. Add 1/4 cup broth. Drain the baking potatoes and arrange half in the Dutch oven in an overlapping layer. Spoon half of the white sauce over the potatoes and sprinkle with a third of the cheese. Arrange the remaining baking potatoes in a second layer, top with the remaining white sauce and sprinkle with the remaining cheese. Cover with the sweet potato slices, then gently press down on the layers. Drizzle the remaining 1/4 cup broth over the sweet potatoes and season with salt and pepper.
Tear a sheet of foil slightly larger than the Dutch oven and butter one side. Place the foil, butter-side down, on top of the sweet potatoes. Place the lid on the Dutch oven and cook over medium heat for 10 minutes. Lower the heat to low and cook until the potatoes are fork-tender, 45 to 50 minutes. Remove from the heat and let stand, covered, for 15 minutes before serving.