Cut each breast half into 2 cutlets. Pound each to 1/8 inch thick.
Mix salt with 3 tbsp flour and coat chicken with this mixture.
In skillet, melt 3 tbsp butter, add cutlets a few at a time and saute until lightly browned on both sides, adding more butter if necessary.
Remove cutlets from the skillet.
Arrange cheese slices on 6 cutlets; top with remaining cutlets and skewer layers together with toothpicks. Set aside.
In the drippings, melt 2 tbsp more butter. Add mushrooms and cook until tender, stirring occasionally. Stir in pepper and 1 tbsp flour; blend well. Gradually stir in water, milk, wine, and bouillon. Heat to boiling, stirring to loosen brown bits from bottom of skillet.
Return cutlets to skillet; heat to boiling. Reduce heat to low; cover and simmer for 5 minutes or until cheese is melted and chicken is thoroughly cooked. Carefully remove toothpicks.
Spoon cutlets and sauce onto a large platter; garnish with parsley sprigs.