Preheat oven to 190C. Lightly grease a medium baking dish.
Place Parmesan cheese, eggs and breadcrumbs in three separate small bowls. Heat the oil in a large frypan over medium-high heat. Dip each piece of chicken into the Parmesan cheese, then into the egg, then into the breadcrumbs. Brown the chicken on both sides in the hot frypan and then transfer them to the prepared baking dish.
Pour wine into frypan and scrape up the browned bits. Add mushrooms and cook for 5 minutes or until tender. Top each chicken breast with even amounts of cheese then spoon mushrooms over the cheese. Pour the remaining wine from the frypan over all. Cover dish with aluminium foil.
Bake 30 to 35 minutes in the preheated oven or until chicken is no longer pink and juices run clear.