Season the chicken cutlets with salt and pepper. Heat a skillet over medium heat and add the olive oil. Brown the chicken on both sides, remove to a plate and set aside. Do not clean the skillet.
Add the butter to the skillet with the mushrooms and shallots. Sauté until the mushrooms are very soft. Add the garlic and the thyme, and continue to sauté for a minute or two. Add the flour, stirring constantly for a couple of minutes to cook the raw taste off.
Add ⅓ of the champagne and stir constantly, scraping up any browned bits from the bottom of the skillet. Add the rest of the champagne and cook while the champagne is thick enough to leave a trail on the bottom of the skillet when a wooden spoon is dragged through it. Stir in the chicken stock and the cream. Taste for salt and pepper and adjust if necessary.
Add the chicken and turn the sauce to low. Cook for 3 or 4 minutes, occasionally spooning sauce on top of the chicken, until the chicken is cooked through. Sprinkle with parmesan and serve
