Heat a heavy based frying pan over a medium heat, and fry the mushrooms in a knob of butter until golden. Transfer the mushrooms to a dish and set aside.
Add another knob of butter and a glug of oil to the frying pan, over a medium-heat heat, season the chicken breasts and fry for about 5 minutes on each side, until golden and cooked through (cooking time will vary depending on the thickness of chicken). Transfer to a wooden board and allow to rest while you make the sauce.
Add the mushrooms and chicken stock to the pan and deglaze over a high heat. With a wooden spoon, scrap the bottom of the pan as it bubbles—this will enrich the sauce with browned, caramelised chicken.
Add the cream, tarragon and season with salt and pepper. Allow to bubble and reduce to your desired sauce consistency—the longer you leave it reduce, the thicker it will get.
Adjust seasoning to your taste and serve alongside the rested chicken breast.
To finish the dish, a squeeze of lemon could work quite nicely here.