Prepare chicken: butterfly the chicken breasts and cut each one into two cutlets.  Working one at a time, place each chicken cutlet between 2 sheets of heavy-duty plastic wrap or a plastic freezer bag, and using a mallet, flatten each cutlet to 1/4 inch thickness. 

Salt and pepper both sides of each chicken cutlet. Add the all-purpose flour to a bowl or plate and dredge each chicken cutlet lightly in flour on both sides.

Allow the chicken to rest while you cook the mushrooms. 

Mushrooms: add one tablespoon of butter to a large pan over medium heat. When the butter is melted and bubbly, add the sliced mushrooms and cook for about 5 to 7 minutes until the mushrooms are softening. Remove the mushrooms to a bowl to be used later. 

Chicken: add a tablespoon of butter to the large skillet that you used to cook the mushrooms. When the butter is melted, add two of the flour-dusted cutlets to the pan. Cook over medium heat for 3 minutes on each side or until the exterior is lightly browned. The goal is to get a little color on the chicken. You do not have to cook it all the way through at this stage because it will go in the oven to finish cooking. Cook the cutlets in two batches, adding 1 tablespoon of butter to the pan before each batch. When a batch of cutlets has completed cooking, move them to a 13 x 9 baking dish and arrange each cutlet in one layer on the bottom of the pan. Do not wipe out or clean the pan between batches.

Once all four of your cutlets are in the baking pan, sprinkle the reserved mushrooms over the chicken. 

Marsala sauce: add the Marsala wine and chicken broth to the reserved pan drippings in the skillet.  Bring to a boil; reduce the heat, and simmer, uncovered, for about 8 minutes, stirring occasionally.  Add salt and pepper, and make sure to scrape the bottom of the pan to loosen any residual chicken fond into the sauce. 

Preheat the oven to 375 F (190 C). 

After the Marsala sauce has reduced a little, pour the sauce evenly over the chicken and mushrooms arranged in the baking dish. 

Combine Fontina and Parmesan cheeses and green onions into a small bowl, and sprinkle evenly over the chicken and mushrooms. 

Bake, uncovered, for 15 minutes. 

Sandwich assembly: slice your bread rolls and add whatever condiments you would like to the bottom part of the roll.

Using a spatula, move each chicken cutlet, covered with melty cheese, on top of the bottom part of each roll you plan to serve. 

Place both the chicken side of the bread roll and the top of the roll open-faced on a sheet pan and place under a broiler set on high for 1 to two minutes. This will toast up the inside top of the roll as well as brown the chicken and cheese, and make sure everything is nice and warm. If you do not have a broiler, you can keep the oven at the temp it was when you baked the chicken Lombardy and cook in the oven for about 5 minutes to toast the bread lightly. 

Remove the sheet pan from the oven and drizzle some of the leftover Marsala sauce from the baking dish onto the cut side of the inside top of each roll to add a bit of moisture to the finished sandwich. Close the rolls and serve.