Clean the mushrooms with kitchen paper to remove any soil and slice.
Peel and finely chop the shallots. Remove the tarragon fronds from the stem and chop finely.
Bring the chicken to room temperature 30 minutes before cooking, pat dry and season with the salt and pepper.
Add the butter and oil to the pan over a medium to high heat.
When the butter has melted add the chicken skin side down and cook for a couple of minutes to get the skin to crisp and start to develop a golden colour. Reduce the heat down to medium for another 8 minutes.
Turn the chicken over and cook for another 5 minutes or until the juices run clear.
Transfer to a clean plate and cover with foil to rest for 5 minutes.
Put the oil and butter in a pan over a medium heat.
Add the shallots and cook for about 2 minutes until soft but not coloured. Stir occasionally with a wooden spoon.
Add the mushrooms and cook for about 5 minutes until soft, stirring occasionally.
Pour in the stock and cook for a further 5 minutes to reduce the liquid.
Add the cream and half of the tarragon and simmer until the sauce has thickened.
Season to taste.
Put the chicken back into the sauce and scatter over the remaining tarragon to serve.