Cut the chicken breasts in half lengthways to create 2 thin pieces. Pound the chicken pieces to 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 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.