Cut each chicken breast in half lengthwise. Place chicken between 2 sheets of heavy-duty plastic wrap or wax paper; flatten to 1/4 - 1/8 inch thickness, using a meat mallet or rolling pin. Season with salt and pepper and put in the fridge for a few hours if possible.
Cook the onions, mushrooms and garlic in 2 tablespoons butter in a large skillet over medium-high heat, stirring constantly, 3 to 5 minutes or just until tender. Remove from the skillet and set aside.
Dredge the chicken pieces in flour.
Add olive oil to a hot skillet and cook the chicken pieces in batches over medium heat 3 to 4 minutes on each side or until golden.
Place chicken in a lightly greased 13 x 9 inch baking dish, overlapping edges. Repeat the procedure with remaining chicken, adding more olive oil if needed.
Deglaze the skillet with the wine and broth and pour over chicken. Spread the onions, mushrooms and garlic evenly over the chicken pieces.
Sprinkle the chicken pieces in this order with the mozzarella, Italian seasoning, Parmesan, onion greens or chives, and breadcrumbs.
Bake uncovered, at 450 degrees for 12 to 14 minutes until cheese melts.