Cut each chicken breast in half length-wise. (Don't worry if they are perfectly even because you will pound them to be equal size) Place each chicken piece between two pieces of plastic wrap and pound lightly with the flat side of a meat mallet (or use something heavy in your kitchen like a rolling pin or can opener) until 1/4 inch thick. Pounding the chicken will tenderize it and reduce the cooking time. Remove the plastic wrap and sprinkle chicken with salt and pepper.
In a large skillet heat oil over medium-high heat. Add the chicken and cook for 3 minutes and then flip and cook for an additional 3 minutes, or until no longer pink. Transfer chicken to a plate and cover to keep warm.
Add 1 teaspoon additional olive oil to the skillet and onion and sliced mushrooms. Sauté until golden and tender. Add broth to skillet. Bring to a boil, stirring to scrape the browned bits off the bottom of the skillet. Boil gently for 10 minutes or until the liquid is reduced to 1/4 cup. Reduce heat to medium low.
Stir in the whipping cream and then add the butter. Sauce should be thickened. Add additional salt and pepper to taste.