Preheat oven to 450°.
Tear off 4 (12- x 18-inch) foil sheets.
Toss together chicken, mushroom slices, and sauce mix; place one-fourth of mixture in center of each foil sheet. Top evenly with tomatoes.
Bring up 2 sides of each foil sheet, and double fold with about 1-inch-wide folds. Double fold each end to form a packet, leaving room for heat circulation inside packet. Place packets on a baking sheet.
Bake at 450° for 15 minutes.
Cook polenta according to package directions.
Open foil packets carefully, allowing steam to escape; spoon ragoût over polenta. Sprinkle with Parmesan cheese.