Preheat your oven to 400 degrees F. Set aside a 9-inch x 13-inch baking dish.
Place a large skillet over medium heat and melt 1 tablespoon of butter in the pan. Add the garlic and mushroom. Cook until the mushrooms have browned and lost most of their water. Transfer the mushrooms to the baking dish and spread them over the bottom.
In a shallow bowl, mix together the flour, salt, and oregano. Coat the chicken breasts all over in the flour mixture.
Add 2 tablespoons of butter to the same skillet, over medium heat. Add the chicken and cook for 4-6 minutes per side until golden brown. (see note)
Pour the Marsala wine and chicken broth into the skillet and to a boil. Season the sauce with salt and pepper to taste.
Place the chicken on top of the mushrooms in the baking dish and pour the sauce over the top. Spread the shredded mozzarella over the chicken and sprinkle with 2/3 of the sliced green onions. Cover the dish with foil.
Bake on the middle oven rack for 10-15 minutes. Carefully remove the foil and bake for another 5-10 minutes until the chicken is cooked through (165 degrees F internal temp) and the cheese is melted.
Remove from the oven and sprinkle with the remaining green onions. Serve warm.
Notes
You may need to cook the chicken in batches so they're not overcrowded and create a lot of liquid in the pan. The chicken should get seared and partially cooked before it finishes cooking in the oven.
If you want the cheese to get golden brown on top, cook uncovered right away or broil after the baking time has finished for 3-4 minutes.