Melt the butter with the oil in a large skillet over quite high heat. Sprinkle the chicken pieces with salt and pepper and fry in batches until well browned, about 5 minutes per side.
Put all the chicken back in the pan, add the chicken stock and reduce the heat to medium. Cover and cook until tender, about 30 minutes. Remove the chicken to a plate and keep warm. Reduce any leftover juices until sticky. Add the wine and the shallots and reduce to a thickish sauce, about 5 minutes. Add the creme fraiche and half the tarragon. Boil down again to sauce consistency, 3 to 5 minutes.
Season the sauce with salt, pepper and lemon juice. Put back the chicken pieces, turning to coat, then transfer to a platter. Pour the sauce over, scatter over the remaining tarragon and serve.