Bring the water, 1 tablespoon butter, and 1/2 teaspoon salt to a boil in a heavy medium saucepan. Gradually whisk in the polenta. Reduce the heat to medium-low. Stir constantly until the polenta thickens, about 5 minutes. Pour the polenta into a greased 9 by 9-inch baking pan, spreading so that it is 1/3-inch thick. Cover and let stand at room temperature until set, about 15 minutes.
Meanwhile, heat the oil in a heavy large frying pan over medium-high heat. Add the mushrooms and onion. Sprinkle with 1/2 teaspoon of salt and 1/4 teaspoon of pepper and saute until the juices evaporate, about 8 minutes. Add the garlic and saute until the mushrooms are golden brown, about 2 minutes longer. Decrease the heat to medium-low. Add the Marsala. Cover and simmer until the Marsala has reduced by about half, about 5 minutes. Stir the flour and remaining 2 tablespoons of butter in a small bowl to form a paste, then stir the paste into the mushroom mixture. Cover and simmer until the sauce thickens slightly, about 2 minutes. Remove from the heat. Stir the parsley into the ragu. Season the ragu, to taste, with more salt and pepper.
Cut the polenta into 36 bite-size squares. Arrange the polenta squares on a platter. Spoon the warm ragu atop the polenta and serve immediately.