Cut the chicken breast into cutlets, (4), sprinkle both sides with salt, pepper, and granulated garlic. Dredge the cutlets in all-purpose flour and shake off the excess.
Heat a large skillet on medium high heat and add 1 tablespoon each of olive oil and butter. Once the oil is shimmering, place 2 of the cutlets in the skillet to brown, 3 to 4 minutes. Turn and brown the second side, 3 to 4 minutes. Remove and place on a plate and tent with foil to keep warm. Add another 1 tablespoon each of olive oil and butter to the skillet and repeat the browning of the other 2 cutlets and place them on the plate tented.
In the same skillet on medium heat, add another tablespoon of butter. Place the sliced mushrooms in the skillet with a pinch of salt and cook until browned and the water from the mushrooms has burned off, 6 to 8 minutes. Add the minced garlic and dried thyme and cook for another minute so that the garlic becomes fragrant.
Pour in the Marsala wine and bring to a boil and reduce by 1/2 which will take 4 to 6 minutes. Next add the chicken broth and bring that to a boil. Once boiling, stir in the flour slurry and bring to a boil to thicken, 2 minutes.
Stir in the heavy cream and place the cutlets back into the skillet to heat up. Spoon the sauce over the cutlets which will further the thickening, 2 or 3 minutes. Taste the sauce and add salt and pepper if needed.
Remove the skillet from the heat and sprinkle with the chopped parsley. Serve warm.