Preheat the slow cooker if necessary; see the manufacturer's instructions. Heat the oil and butter in a frying pan, add the chicken and fry over a high heat until golden on both sides but not cooked through. Drain and put into the slow cooker pot in a single layer.
Add the shallots to the frying pan and cook, stirring, for 4–5 minutes or until just beginning to turn golden. Stir in the flour, then gradually mix in the stock and vermouth. Add the sprigs of tarragon, a little salt and pepper and bring to the boil, stirring.
Pour the sauce over the chicken breasts, cover with the lid and cook on high for 3–4 hours or until the chicken is cooked through to the centre.
Stir the cream into the sauce and sprinkle the chicken with 1 tablespoon chopped tarragon and the chives. Serve with coarsely mashed potatoes mixed with peas.