Slice each chicken breast in half to make two thinner cutlets of even thickness. You should end up with a total of 8 thin pieces of chicken.
In a shallow bowl, combine flour, salt and pepper. In a large skillet, melt butter over medium low heat. Dredge chicken breasts in flour mixture and brown in the pan, 3-4 minutes per side. Make the chicken in batches if necessary. Add the mushrooms, cover and simmer for 10 minutes.
Add the cream and simmer, uncovered, for an additional 10 minutes. Transfer chicken breasts to a warm platter.
Add the champagne to the pan and bring to a rapid boil over medium-high heat, whisking any browned bits off the bottom of the pan. Lower heat and allow sauce to reduce until creamy.
Place the chicken and mushrooms on a serving plate. Pour sauce over the chicken breasts. Garnish with fresh parsley.