01 - In a medium saucepan, bring the water or vegetable broth and whole milk to a rolling boil. Gradually whisk in the polenta and salt, ensuring no lumps form. Reduce the heat to low and continue to cook, stirring frequently, until the polenta becomes thick and creamy, which will take approximately 25-30 minutes.
02 - Stir the unsalted butter, grated Parmesan cheese, and freshly ground black pepper into the cooked polenta. Maintain its warmth over very low heat, stirring occasionally to prevent sticking.
03 - Concurrently, heat the olive oil and unsalted butter in a large skillet over medium heat. Add the finely chopped yellow onion and cook until it becomes soft and translucent, typically about 3 minutes.
04 - Introduce the minced garlic to the skillet and cook for an additional 30 seconds, until fragrant.
05 - Add the sliced mixed mushrooms and fresh thyme leaves to the skillet. Cook, stirring intermittently, until the mushrooms are well browned and have released their liquid, which should take about 7 minutes.
06 - Pour in the dry white wine and allow it to simmer until it has mostly evaporated, approximately 2 minutes.
07 - Add the vegetable broth to the mushroom mixture. Reduce the heat and gently simmer until the sauce is rich and thickened but not overly soupy, about 5 minutes. Season with salt and black pepper to taste.
08 - To serve, spoon the creamy polenta into individual bowls. Generously top with the braised mushrooms and garnish with chopped fresh parsley.