Saute the mushrooms in 2 tablespoons of butter in a large skillet until tender and set aside.
Cut each chicken breast in half lengthwise. Pound until 1/8 inch thick between saran wrap or wax paper.
Coat the chicken with the flour. Melt 2 tablespoons of butter in another large skillet over medium heat. Add the chicken in batches cooking each side for 3 to 4 minutes. Remove each batch and place the chicken in a 9x13 baking dish, over lapping if necessary. Sprinkle the mushrooms over the chicken.
To make the sauce, add the wine and chicken broth to the drippings in the skillet, stirring to deglaze the skillet. Bring to a boil and reduce the heat to simmer for 8 minutes, stirring occasionally. If you need to thicken add a tablespoon of cornstarch and a little chicken broth that is blended before you put it in the sauce. Stir in the salt and pepper, then pour over the chicken.
Combine the fontina, parmesan cheese and green onions in a bowl. Sprinkle over the chicken and bake 20 mins at 375 degrees. Broil for 2 mins 6 inches from the heat source. . Leave the oven door open a bit