Melt the butter with the oil in a large skillet over quite high heat. Season the chicken pieces and fry in batches until well browned, about 5 minutes per side. Put all the chicken back in the pan, add 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 and reduce any leftover juices until sticky. Add the shallot and wine and reduce to a thickish sauce, about 5 minutes. Add the cream 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.