Cut the chicken in half lengthwise to make 4 thinner cutlets. In a shallow dish or plate, combine the flour, salt, and black pepper. Add the chicken and dredge each piece completely.
Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. When it's hot, add the chicken and cook for 3-4 minutes on each side until golden brown. It's OKAY if the chicken is not fully cooked at this stage. We will finish cooking them in the sauce. Remove the chicken to a plate.
Add the remaining 2 tablespoons of oil to the skillet. Add the onions and cook for 3 minutes until softened. Add the garlic and cook for 30 seconds. Add the mushrooms, and cook for 4-5 minutes until most of their moisture has been released.
Add the marsala wine and bring to a boil. Let the mixture bubble for 3-5 minutes until it has reduced in volume by half.
Stir in the heavy cream and butter. Add back the chicken and cook for another 3-4 minutes until the chicken is cooked through and the sauce has thickened slightly. Enjoy!