Cut each chicken breast in half lengthways to create 2 thin pieces, for a total of 6 thin pieces. Pound the chicken pieces with a mallet about ¼ inch thick.
Heat the oil to medium high heat in a deep large skillet.
Place the flour, salt, and pepper onto a plate. Whisk this together with a fork.
Take each piece of chicken one at a time and coat it with the flour mixture. Shake off any excess, then place the pieces into the hot oil. (Cook just a couple at a time so they are not crowded.)
Cook for 3-4 minutes on each side until cooked fully through and seared to a golden brown. Take the chicken pieces from the skillet and place them on a plate. Cover them with tin foil.
Melt butter in the same skillet, then add mushrooms and diced onion and fry for about 3 minutes.
Add the 1 Tablespoon of flour and minced garlic cooking for another minute. Make sure to scrape the bottom of the pan to release any bits stuck there.
Whisk in Marsala wine and simmer it for a minute.
Whisk in chicken broth and heavy cream. Bring this to a boil. Then simmer it to thicken and reduce the sauce, deglazing the pot every so often. This part of the process will take 8 to 10 minutes. You want to give it plenty of time to deepen in color and thicken up.
Return the chicken to the skillet continuing to heat for a few minutes, or until the chicken is heated through.
Notes
Please note that this dish can really be made to order! You can add more mushrooms and onions and even double the sauce recipe if you are like me and like your food drowning in sauce! There is plenty of sauce for 6 pieces, but I love extra!