Clean the mushrooms with a brush; do not wash with water because it will dilute the flavor. Trim the mushrooms, slice large caps into fairly even pieces and leave smaller caps intact. 

In a fry pan over medium heat, warm the olive oil. Add the mushrooms and cook, stirring occasionally, until lightly browned and all of the moisture has evaporated. Add the onion and garlic and cook, stirring occasionally, until translucent. Add the tomato paste and cook, stirring occasionally, for 3 to 4 minutes. Add the wine and deglaze the pan, stirring to scrape up the browned bits, then cook until all of the alcohol has evaporated and the liquid is reduced by half. Add the stock and bay leaves. Simmer until the mixture reaches a sauce-like consistency and coats the back of a spoon. 

Remove the pan from the heat and add the butter, thyme and parsley. Season with salt and pepper and stir until the butter has melted. Serve over the creamy polenta (recipe below) and garnish with shaved Parmesan.

In a large saucepan over medium heat, melt the 2 tablespoons butter with the olive oil. Add the onion and cook, stirring occasionally, until lightly translucent but not browned. Stir in the rosemary. 

Add the stock and milk and bring to a boil over medium-high heat, then reduce the heat to a simmer. Slowly add the polenta, whisking constantly to prevent lumps from forming. Simmer for 20 minutes, stirring constantly to prevent the polenta from burning and sticking to the bottom of the pan. 

When the polenta has developed a creamy consistency, season with salt and pepper. Add the cheese and the 1/2 cup butter and mix well. If the polenta becomes too thick for your liking, add more stock or milk to thin it out a bit. Serve hot.