Bring the milk to the boil in a pan. Stir in the polenta, reduce the heat, simmer for approx. 45 mins. to create a thick paste, stirring frequently. Stir in the cheese, season. Spread the polenta (approx. 1.5 cm thick) on a baking tray lined with baking paper, leave to cool ca. 2 hours. Cut out approx. 6 cm circles.
Heat the oil in a pan, sauté the onion, garlic and vegetables. Add the tomato puree, bay leaf and thyme, briefly sauté. Pour in the wine and stock, bring to the boil, then reduce the heat and simmer the vegetables for approx. 20 mins. Heat the butter in a frying pan, briefly sauté the mushrooms, add to the vegetables, simmer for approx. 5 mins. Stir in the parsley, season.
Transfer the mushroom ragout to a greased, ovenproof dish (approx. 3 l). Spread the polenta rounds on top, sprinkle with cheese.
Gratinate for approx. 10 mins. in the centre of an oven preheated to 200°C.