1. heat the oven to 350 degrees. In a medium sauce pan 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 oil and ¼ 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 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, ¼ teaspoon of 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, ¼ teaspoon salt and sage and 1/8 teaspoon black 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.