Adjust the oven rack to the middle position and heat the oven to 350 degrees. Melt the butter in a Dutch oven over medium-high heat until foaming. Add the shallots and half of the salt and cook until softened, about 2 minutes. Stir in the chard and cook until wilted, about 2 minutes. Add the garlic, thyme, and pepper and cook until fragrant. Transfer the mixture to a bowl.

In the same empty pot, combine the cream, water, wine, and the remaining salt. Bring the mixture to a simmer over medium-high heat. Once simmering, remove the pot from the heat and cover to keep the liquid warm.

Layer half of the sliced potatoes evenly into a 3-quart gratin dish. Spread the wilted chard mixture over the potatoes, then top with the remaining potatoes. Pour the hot cream mixture evenly over the top.

Cover the dish tightly with aluminum foil and bake for 45 minutes. Remove the foil and sprinkle the Parmesan evenly over the potatoes.

Continue to bake uncovered until the top is deeply golden brown and nearly all the liquid has evaporated, about 45 minutes. Let the gratin rest for 10 minutes before serving to allow the sauce to set.