Step 1In a medium, heavy-bottom saucepan, combine the cream and milk and bring to a simmer. Slowly whisk in the polenta and reduce the heat to low. Cook, stirring every few minutes or so, making sure to scrape the bottom of the pan so the polenta does not burn. Cook the polenta until it is creamy, about 30 minutes, adding additional milk if the polenta thickens too quickly before it is tender. Remove from heat, cover and set aside for 20 to 30 minutes.
Step 2Whisk in the butter and grated cheese, then taste and adjust the seasonings with salt and pepper.
Step 1In a large saute pan, heat the oil, butter and herbs over medium heat, cooking until the herbs are crisp and the butter begins to brown, 4 to 5 minutes.
Step 2Add the mushrooms and season with 1 teaspoon salt and a few grinds of pepper, or to taste. Toss, coating the mushrooms with the fat and increase the heat to high, cooking until the mushrooms brown a little, 3 to 4 minutes. Reduce the heat to medium.
Step 3Make a well in the center of the pan and add the garlic and chile flakes. Stir the garlic until it is cooked through, 1 to 2 minutes, then combine it with the mushrooms. Add the wine and broth, then remove from heat and add the vinegar. Taste, adjusting the seasonings as desired.
Step 4To serve, fill a serving dish with polenta, then top with the mushrooms.