Skin the chicken breasts and slice the meat vertically, across the grain of meat, into even-sized thin escalopes of about a centimeter in thickness, to give roughly 3 pieces per breast. Season with salt and pepper.
Heat a large frying pan over a medium heat and add the oil. Add the chicken piece one at a time, ensuring they are not on top of each other. Leave them alone in the pan to allow a nice crust to form. Turn the chicken pieces over and cook for another 2-3 minutes as above.
Add the butter, the soaked morels (retain the liquid for later) and quartered chestnut mushrooms to the pan, moving them around in the butter to ensure they take on a nice colour. Pour in the wine and deglaze using a heatproof spatula to scrape all the flavour and caramelised pieces from the bottom of the pan. Reduce by half, then add the stick plus 4 tablespoons of the strained mushroom soaking liquid and reduce by half again.
Add the cream and chopped tarragon and bring to a simmer. Further reduce he sauce until it has a good coating consistency on the back of the spoon. If you have a thermoprobe thermometer, the serving temperature of the meat should be 80°C. Check the seasoning.
For the potatoes, boil them in plenty of boiling, salted water until tender – for 15-20minutes, then lightly fork them with the butter, pepper and chopped parsley. When you’re ready to serve, spoon the hot crushed parsley potato into a mousse ring (or cake cutter) on a warmed plate to make a neat round helping. Pat down gently then remove the mousse ring. Place the chicken on the plates with steamed greens on the side and the sauce and mushrooms spooned round.