1. Splash a little water over both sides of each chicken breast half and place smooth side down on separate sheets of waxed paper.
2. Fold out the fillet (extra flap) and top with additional waxed paper. Start ing at the center gently pound the breasts until they are between 1/8 and 1/4 i nch thick. Peel off the top waxed paper and lift the pounded cutlet from the w ax paper underneath so it does not stick.
3. Put 1 Tbs. each of the butter and olive oil in a large skillet over medium h eat. Sprinkle the flour on a plate, and dredge 2 cutlets to lightly coat. Pla ce in heated butter and brown well on each side about 2 minutes; set aside cove red with aluminum foil.
4. Repeat with remaining 2 cutlets.
5. Add the remaining 1/2 Tbs. butter to the pan with the garlic and rosemary; saute for 30 seconds and then add the mushrooms; brown well 1-2 minutes.
6. Pour in the Marsala and add the salt and pepper; boil for about 1 minute. Return all the cutlets in a single layer (slightly overlapping is ok) to the pa n.
7. Top each cutlet with a slice if ham and cheese, trimming to fit each breast . Cover the pan and bring to a boil over moderate heat, lower heat and simmer until cheese melts (about 2 minutes).
8. Remove the cutlets from the pan with a spatula and transfer to serving palt es, spoon mushroom sauce over top and sprinkle with parsely.