01 - Bring water or broth and milk to a boil in a medium saucepan. Gradually whisk in polenta and salt. Reduce heat to low and cook, stirring frequently, until thick and creamy, about 25-30 minutes.
02 - Stir butter, Parmesan cheese, and black pepper into the polenta until melted and incorporated. Keep warm over very low heat, stirring occasionally to prevent sticking.
03 - Heat olive oil and butter in a large skillet over medium heat. Add chopped onion and cook until soft and translucent, about 3 minutes.
04 - Add minced garlic to the skillet and cook for 30 seconds until fragrant, being careful not to burn it.
05 - Add sliced mushrooms and thyme. Cook, stirring occasionally, until mushrooms are browned and have released their liquid, about 7 minutes.
06 - Pour in white wine and let it simmer until mostly evaporated, about 2 minutes, allowing the alcohol to cook off and flavors to concentrate.
07 - Add vegetable broth and reduce heat. Simmer until the mixture reaches a saucy consistency, about 5 minutes. Season with salt and pepper to taste.
08 - Spoon creamy polenta into serving bowls. Top generously with braised mushrooms and their sauce. Garnish with chopped fresh parsley.