Combine milk and water in a medium saucepan. Bring to a simmer. Pour in polenta in a thin stream, whisking constantly until it is all added. Simmer over medium-low heat 20 minutes or until thick.
Meanwhile, heat oil in a large skillet over medium heat until shimmering. Add onion and saute until limp, 3-5 minutes. Add garlic and mushrooms and saute until mushrooms are tender, 4-5 minutes. Add the can of tomatoes (with juices) and red pepper flakes. Bring to a boil, reduce to a simmer, and cook, uncovered, 10 minutes. Season with salt and pepper.
Add butter and cheese to polenta off-heat and stir well. Divide polenta between two serving plates. Top each with half the mushrooms and serve hot.