Cut each chicken breast evenly in half, lengthwise. One at a time, place a piece of chicken in between two sheets of heavy duty plastic wrap. Using the flat side of a meat mallet, flatten the breast out to a 1/4" thickness. Repeat for all the pieces of chicken.
Melt the two tablespoons of butter in a large skillet over medium heat. Add in the mushrooms and saute for 3-5 minutes, or until tender. Remove the cooked mushrooms to a waiting plate or bowl, and set aside.
Add the flour to a flat plate. Dredge each piece of chicken in it, making sure there's an even coat.
In the same skillet you cooked the mushrooms in, melt two tablespoons of the remaining butter. Add two pieces of chicken, and up the heat to medium high. Let the chicken cook until nicely browned on one side, flip and repeat. Transfer the browned breasts to a waiting plate.
Add another tablespoon of butter to the skillet, and brown another two breasts. Repeat the steps until all the chicken is browned, and all transferred to the waiting plate.
Stir the broth and wine into the drippings in the skillet, whisking to deglaze the pan and get up all those yummy browned bits. Season with salt & pepper. Bring the mixture to a boil, then reduce the heat and simmer for 5 minutes, or until thickened slightly.
In a small bowl, whisk together the corn starch and water to make a slurry. Whisk this into the skillet sauce, and let it simmer another 1-2 minutes- until nicely thickened. Remove the skillet from heat, and set aside.
Transfer the chicken breasts to a lightly greased 9x13 baking dish, overlapping a bit if necessary to fit them all. Spread the sauteed mushrooms evenly out over top.
Pour the sauce evenly out over top of the chicken. Then sprinkle the cheeses evenly out over top, followed by the sliced green onions.
Bake the prepared chicken at 450 degrees for 15-20 minutes, or until the cheese is melted and slightly golden brown. Remove the dish from the oven, and serve immediately.