Prepare the chicken breasts by pounding with a tenderizer to ensure even in size. Slice each chicken breast in half, to make six servings. Lightly season both sides with desired amount of salt and pepper.
In a large skillet, heat 2 Tablespoons of oil over medium heat. Add chicken to skillet and cook for about 5 minutes on each side or until golden brown on both sides. Remove from skillet and set aside.
In the same skillet, add the remaining 2 Tablespoons of olive oil. Cook the shallots and mushrooms together until tender. Add garlic and cook for one additional minute.
Pour in the champagne. Bring to a boil, then reduce heat to a simmer and cook for about 10 minutes.
Add the milk and stir to combine, then place the chicken back in the skillet and cover with sauce. Continue to simmer 10 minutes or until chicken is cooked through.
Mix cornstarch with an equal amount of water, and add to skillet. When sauce begins to thicken, remove from heat, top with fresh parsley and serve.