How to Make It
In a large saucepan, combine 4 1/2 cups of the stock with the milk and bring to a boil. Whisk in the polenta in a slow, steady stream. Add a generous pinch of salt and cook over moderately low heat, whisking frequently, until the polenta pulls away from the side of the pot and it's thick and creamy, about 30 minutes. Whisk in 4 tablespoons of the butter and the 1/4 cup of cheese and season with salt and pepper. Remove from the heat. Press plastic wrap directly onto the surface of the polenta to prevent a film from forming. Cover and keep warm, adding tablespoons of water if it gets too thick.
In a large skillet, heat 2 tablespoons of the olive oil until shimmering. Add half of the mushrooms and season with salt and pepper. Cook over moderately high heat, stirring occasionally, until the mushrooms are golden and tender, about 5 minutes; transfer to a plate. Repeat with the remaining 2 tablespoons of olive oil and mushrooms.
Return all of the mushrooms to the skillet. Add the vinegar and lemon juice and cook over moderate heat, stirring, until absorbed, about 2 minutes. Add the thyme, rosemary and the remaining 1 cup of stock and cook, stirring occasionally, until the mushrooms are coated in a light sauce, 3 to 5 minutes. Stir in the remaining 2 tablespoons of butter and 2 tablespoons of parsley, then season with salt and pepper.
Transfer the polenta to shallow bowls and spoon the mushrooms on top. Garnish with chopped parsley and serve, passing grated cheese at the table.