Cut each chicken-breast equally in half, lengthwise. One at a time, put a sheet of chicken between 2 sheets of heavy duty plastic-wrap. Together with the flat side of a meat mallet, flatten out the breast into some 1/4" thickness. Repeat for several of the bits of chicken.
Melt both 2 tbsp of butter in a large skillet over moderate heat. Add the mushrooms and simmer for 3-5 mins, or till tender. Remove the cooked mushrooms into a waiting bowl or plate and set aside.
Add the flour to a plate. Dredge each piece of chicken inside it, making sure that there's an even coating.
At precisely the exact same skillet you cooked the mushrooms melt 2 tbsp of the remaining butter. Insert 2 pieces chicken & also the heat to medium large. Allow the chicken cook until well browned on one side, reverse & repeat. Transfer the browned breasts into a waiting plate.
Add another tbsp of butter into the skillet & brown the following 2 breasts. Repeat the following steps until all of the chicken is browned, and all moved into the waiting plate.
Stir the wine and broth to the drippings from the skilletto deglaze the pan and then get up those yummy browned pieces. Season with salt & pepper. Bring the mixture to a boil, then lower the heat and simmer for 5 mins, or until thickened slightly.
In a small-bowl, whisk together the corn-starch and water to produce a slurry. Whisk this into the skillet sauce, allow it to simmer another 1-2 mins till well thickened. Remove the skillet from heat, and set aside.
Transfer the chicken breasts into a lightly greased 9x13 baking dish, overlapping somewhat if required to match all of them. Distribute the sauteed-mushrooms evenly out along the top.
Pour the sauce evenly out on top of the chicken. Then sprinkle the cheeses evenly out on the top, followed with the chopped green-onions.
Bake the ready chicken in 450 for 15-20 mins, or until the cheese is melted and slightly golden-brown. Remove the dish from the oven & serve immediately.