1. Heat the oven to 350°. In a medium saucepan, bring the water and 1 teaspoon of the salt to a boil. Add the cornmeal in a slow stream, whisking. Whisk in 1 tablespoon of the oil and 1/4 teaspoon of the sage. Reduce the heat and simmer, stirring frequently with a wooden spoon, until very thick, about 20 minutes. Stir in 3 tablespoons of the Parmesan.
2. Meanwhile, butter an 8-by-12-inch baking dish. In a large frying pan, melt 1 tablespoon of the butter with 1 tablespoon of the oil over moderately high heat. Add half the mushrooms, 1/4 teaspoon each of the salt and sage, and 1/8 teaspoon of the pepper. Cook, stirring frequently, until the mushrooms are golden, about 5 minutes. Remove. Repeat with the remaining mushrooms, 1 tablespoon each butter and oil, 1/4 teaspoon each salt and sage, and 1/8 teaspoon pepper.
3. Pour half the polenta into the baking dish and spread in an even layer. Top with half the mushrooms, followed by half of the fontina and 2 tablespoons of the Parmesan. Repeat with the remaining polenta, mushrooms, fontina, and Parmesan. Bake until the cheese is bubbling, about 15 minutes.
Wine Recommendation: To contrast with the cheese here, look for a wine with both good acidity and well-delineated fruit flavors. The earthy red-berry flavors and refreshing tartness of a Bourgogne Rouge, the basic red wine of Burgundy, will be perfect.