Preheat the oven to 350°F/180°C. Prepare a baking dish large enough to hold the chicken with cooking spray.
Saute the onions in a small amount of olive oil until softened and beginning to change color. Add the mushrooms and cook until soft. Season with salt, pepper and thyme. Set aside.
Meanwhile, pound the sliced chicken breasts flat by placing a piece of heavy duty plastic wrap over top and pounding it 1/2" thick with a mallet.
Dredge the chicken breasts in the seasoned flour.
Heat the olive oil and butter in a large skillet and fry the chicken breasts for three or four minutes on each side until golden brown. Work in batches as needed.
Transfer the chicken to the baking dish. It's okay if the pieces overlap.
Using the same skillet, make the sauce by adding the chicken broth and marsala wine and stirring up any brown bits from the bottom of the pan. Bring the broth to a boil and allow it to reduce slightly. If you want, add a knob of butter to make the sauce even richer.
Distribute the onions and mushrooms over the chicken and pour on the sauce.
Spread the cheese over top and bake for 15 - 20 minutes or until the cheese is melted.