Adjust oven rack to middle position and heat oven to 400ºF. Bring water and milk to boil in large saucepan over medium-high heat. Stir in 1 teaspoon salt. Slowly pour polenta into liquid in steady stream while stirring back and forth with wooden spoon. Reduce to gentle simmer, cover, and cook, stirring often, until mixture has uniformly smooth, thick consistency, 15 to 20 minutes. Off heat, stir in Parmesan and butter, and season with salt and pepper to taste. Pour polenta into 13 by 9-inch baking dish and smooth into even layer.
Meanwhile, heat oil in 12-inch skillet over medium heat until shimmering. Add onion and ½ teaspoon salt and cook until onion is softened, about 5 minutes. Add mushrooms and cook until they have released their liquid and are well browned, about 25 minutes.
Stir in garlic and thyme and cook until fragrant, about 30 seconds. Stir in tomatoes and their juice, bring to simmer, and cook, stirring occasionally, until sauce has thickened, about 10 minutes. Stir in chard, 1 handful at a time, and cook until wilted, 2 to 4 minutes. Season with salt and pepper to taste.
Spread mushroom mixture evenly over polenta, then sprinkle with fontina. Bake casserole until warmed through and cheese is melted, 10 to 15 minutes. Let cool for 5 minutes before serving.