How to Make It
Heat the oven to 350°. Oil an 8-by-12-inch baking dish. In a medium saucepan, bring the water and 2 teaspoons of the salt to a boil. Add the cornmeal in a slow stream, whisking constantly. Reduce the heat to moderate. Simmer, stirring frequently with a wooden spoon, until the polenta is very thick and pulls away from the sides of the pan, about 20 minutes. Stir in 4 1/2 tablespoons of the butter and 1/2 cup of the Parmesan. Spread the polenta in the prepared dish.
If using shiitakes or portobellos, remove the stems. Slice all the mushrooms. In a large frying pan, melt the remaining 1 tablespoon butter over moderately low heat. Add the shallot and cook, stirring, until translucent, about 3 minutes. Add the garlic and cook, stirring, until fragrant, about 30 seconds. Add the mushrooms, sage and remaining teaspoon salt and cook, stirring occasionally, until the mushrooms are brown, about 5 minutes. Add the pepper. Pour the mixture over the polenta and sprinkle with the remaining cheese. Bake until lightly browned, about 10 minutes.