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 brown bits off the bottom of the pan. Lower heat and allow sauce to reduce until creamy.
Bring a large pot of salted water to a boil. Peel and quarter the potatoes. Boil potatoes 10-15 minutes, until potatoes are easily pierced with a fork. Drain. Over low heat, melt the butter in the same pot. Add the potatoes, milk and spices. Remove from heat. Using an electric mixer, whip the potatoes until smooth.
Spread the potatoes onto a serving dish. Place the chicken and mushrooms on top of the potatoes. Spoon the sauce over chicken breasts. Garnish with fresh parsley.