Preheat a grill or broiler. If using a grill, also preheat the oven to 450°F.
Place a saucepan of water over medium high heat and bring to a boil.
Add broccoli. Cook just until bright green and tender-crisp.
Immediately plunge broccoli into ice water to stop the cooking.
Drain and pat dry.
Use a meat mallet, flatten chicken to a uniform thickness of 1/2 to 3/4-inch.
Place flattened chicken on a plate and coat with olive oil.
Place bread crumbs into a large shallow bowl.
Dredge both sides of each chicken breast in bread crumbs.
Place chicken on grill or under broiler and cook about 4 minutes per side, to an internal temperature of 165 degrees. (Insert thermometer horizontally for an accurate reading.) If using the broiler, set oven at 450°F when you remove the chicken.
Pour prepared sauce into a large skillet.
Add prosciutto, red pepper flakes and garlic. Place over medium heat and bring to a boil.
Add broccoli and mushrooms. Continue to cook until sauce thickens and reduces by about 1/3.
In a small bowl, mix cheeses together.
Place cooked chicken on a shallow baking pan.
Top each piece with 1/2 cup combined cheeses.
Place in preheated oven and bake until cheese is melted, about 4 minutes.
Transfer chicken to serving plates.
Use a slotted spoon to evenly distribute broccoli and mushrooms over chicken.
Pour on remaining sauce, to taste.
Serve immediately.