Bring milk, water, polenta, salt, and pepper to boil in heavy large saucepan over medium heat, whisking constantly. Reduce heat to low and simmer until thick, stirring occasionally, about 20 minutes. Remove from heat.

Meanwhile, saut� onion in 2 tablespoons oil in a heavy large skillet over medium-high heat until golden brown, about 3 minutes. Add mushrooms to skillet. Saut� until mushrooms are tender, about 6 minutes.

Season to taste with salt and pepper.

Whisk butter and Parmesan into polenta and divide among plates. Top with onion and mushroom mixture.