Adjust an oven rack to the middle position and heat the oven to 400°F. Grease a 13x9-inch baking dish.

Melt the butter in a large Dutch oven over medium heat. Add the shallots and a pinch of salt and cook until they are softened and translucent, about 3 minutes. Stir in the chard, garlic, and thyme and cook until the greens are completely wilted and any released liquid has evaporated, 5 to 7 minutes.

Whisk the cream, water, wine, salt, and pepper in a medium bowl or large liquid measuring cup until combined.

Arrange half of the sweet potato slices in an even layer in the bottom of the prepared baking dish. Spread the chard mixture uniformly over the potatoes, then shingle the remaining potato slices on top. Pour the cream mixture over the vegetables.

Cover the dish tightly with aluminum foil and bake for 45 minutes—the steam trapped inside will soften the dense potatoes. The potatoes should be just tender when pierced with a paring knife.

Remove the foil and sprinkle the Parmesan evenly over the surface. Bake uncovered until the sauce has thickened and the cheese is bubbly and deeply golden brown, 20 to 30 minutes.

Let the gratin rest for 10 minutes before serving so the starches can settle, which makes it much easier to cut into clean portions.