Adjust an oven rack to the middle position and heat the oven to 400°F. Bring the water and milk to a boil in a large saucepan over medium-high heat. Stir in the salt, then slowly pour the polenta into the liquid in a steady stream while stirring back and forth with a wooden spoon—this prevents lumps from forming. Reduce to a gentle simmer, cover, and cook until the mixture is uniformly smooth and thick, 15 to 20 minutes, stirring often. Off the heat, stir in the Parmesan and butter, season with salt and pepper, and spread the mixture into an even layer in a 13 by 9-inch baking dish.

Heat the oil in a 12-inch skillet over medium heat until shimmering. Add the onion and salt and cook until softened, about 5 minutes. Stir in the mushrooms and cook until they have released their liquid and are well-browned, about 25 minutes; this deep color is essential for a savory base.

Add the garlic and thyme and cook until fragrant. Stir in the tomatoes and their juice, bring to a simmer, and cook until the sauce has thickened, about 10 minutes. Add the chard one handful at a time, stirring until wilted, which should take 2 to 4 minutes. Season the mixture with salt and pepper to taste.

Spread the mushroom and chard mixture evenly over the polenta base, then sprinkle with the fontina. Bake until the casserole is warmed through and the cheese has completely melted, 10 to 15 minutes. Let the dish cool for 5 minutes before serving to allow the layers to set.