Bring the broth with the milk to a boil. Stir in the polenta, turn the heat down very low and stir in the semolina for a minute. Put the lid on and let it soak for 15 minutes. Stirring occasionally.
Meanwhile, heat the oil in the pan. Quarter the mushrooms, cut them into small pieces and then fry them in the hot oil for a few minutes. Add the thyme and the tomato puree. Season with salt and pepper, reduce the heat. Put the lid on and let the mushrooms stew for about 10 minutes. After about 8 minutes stir in the wholemeal sauce thickener.
After swelling, stir the cheese into the corn porridge. Put the lid back on and heat for a few more minutes until the cheese has melted. Stir well, season with salt and pepper and serve with the mushrooms.