Lightly flour chicken breasts, shaking off excess flour. In a large frypan over low heat, melt butter. Add chicken and saute until lightly browned, about 2 minutes each side. Remove with slotted spatula and set aside.
Increase heat to medium low. Add mushrooms and saute until juices are rendered, about 4 minutes. Add wine and simmer until reduced by 1/4, about 3 to 4 minutes. Increase heat to medium high. Add stock, parsley and pepper and simmer until sauce reduced to 1 cup, about 10 minutes.
Reduce heat to low. Top each chicken breast with a slice of prosciutto and a slice of cheese. Return chicken to frypan and cook just until cheese melts. Transfer chicken to individual plates and top each with some of the mushroom sauce before serving.