Prepare the mushrooms by cleaning them with a damp paper towel (don’t submerge them in water). Trim stems and slice mushrooms into bite-sized pieces, keeping in mind they’ll shrink during cooking.
Bring the broth to a boil in a heavy-bottomed pot. Once boiling, slowly pour in the polenta while whisking constantly to prevent lumps from forming.
Reduce heat to low and continue cooking the polenta for 20-25 minutes, stirring frequently with a wooden spoon. The mixture will thicken considerably and should pull away from the sides of the pot when ready.
Meanwhile, heat olive oil and butter in a large skillet over medium-high heat until the butter is melted but not browning.
Add the mushrooms in batches (overcrowding prevents proper browning). Cook without stirring for 2-3 minutes to develop caramelization, then stir occasionally until golden brown, about 5-7 minutes total.
Add shallots and garlic to the mushrooms and cook for 1-2 minutes until fragrant but not browned.
Sprinkle in thyme and rosemary, then deglaze the pan with white wine, scraping up any browned bits from the bottom. Cook until liquid is mostly evaporated.
Finish the polenta by stirring in the Parmesan cheese, butter, and cream until fully incorporated. Season with salt and pepper to taste.
Taste and season the mushroom mixture with salt and pepper, then fold in the fresh parsley.
Serve immediately by spooning the creamy polenta into shallow bowls or onto a platter, then topping with the sautéed mushrooms. Garnish with additional Parmesan if desired.