Heat the olive oil in a skillet of pan over medium heat. Season each chicken filet with a pinch of salt. Sear chicken on both sides, for about 3-5 minutes each side (depending on thickness), until golden all over. Transfer chicken to a plate.
Fry the mushrooms in the butter and fry for a further 3-5 minutes, or until just beginning to soften. Transfer to the same plate as the chicken. Cover and keep warm.
Add the shallot into the pan and cook for 4 minutes, while occasionally stirring. Pour in the champagne; stir well, while scraping any food bits from the bottom of the pan for added flavour. Season with salt and pepper and allow to simmer for about 10 minutes.
Add milk (or cream) and simmer for 2-3 minutes. If using milk: mix the cornstarch with a tablespoon of water; pour into the centre of the pan and mix throughout the sauce until slightly thickened.
Add the chicken and mushrooms into the sauce along with any cooking juices. Coat in the creamy sauce. Sprinkle with parsley and serve!