Bring the water and milk to a boil in a large saucepan. Whisk in the polenta and salt, then reduce the heat to low and simmer, stirring occasionally, until the grains are tender and the mixture is thick, about 25 to 30 minutes.

Stir the Parmesan, butter, and pepper into the cooked polenta, then spread the mixture into an even layer in a 13 by 9-inch baking dish.

Heat the olive oil in a large skillet over medium-high heat until it shimmers. Add the mushrooms and cook, stirring only occasionally, until they release their liquid and become deeply browned, about 12 to 15 minutes.

Add the onion to the skillet and cook until softened and translucent. Stir in the garlic and thyme until fragrant, then add the tomatoes and their juice. Simmer the mixture until the sauce has thickened and the liquid is mostly evaporated, about 10 minutes—reducing the liquid here prevents a soggy casserole.

Fold in the Swiss chard and cook just until the leaves wilt and turn bright green.

Spoon the mushroom mixture over the polenta and top with the fontina. Bake at 400°F until the cheese is melted, golden, and bubbling at the edges, about 15 minutes.

Serve hot, letting the casserole sit for 5 minutes to set before slicing.