Bring the water, milk, salt, and pepper to a boil in a large saucepan over medium-high heat. Slowly whisk in the polenta, then reduce the heat to low and simmer, stirring frequently, until the grains are tender and the mixture is thick, about 25 to 30 minutes.

Stir the Parmesan and butter into the cooked polenta until melted, then cover and set aside to stay warm.

Adjust an oven rack to the middle position and heat the oven to 400 degrees. Lightly grease a 9 by 13-inch baking dish.

Heat the olive oil in a large skillet over medium-high heat until shimmering. Add the onion and mushrooms and cook until the mushroom liquid has completely evaporated and the vegetables are deeply browned, 12 to 15 minutes.

Stir in the garlic and thyme and cook until fragrant. Add the tomatoes and their juice and simmer until the sauce has thickened and most of the liquid has evaporated, about 10 minutes.

Stir in the Swiss chard and cook until just wilted, about 3 minutes. Season the mixture with additional salt and pepper to taste.

Spread the warm polenta evenly into the prepared baking dish. Top with the mushroom and chard mixture, then sprinkle the fontina evenly over the top.

Bake until the casserole is hot throughout and the cheese is melted and beginning to brown, about 15 to 20 minutes. Serve hot.