Set a 5L slow cooker to Browning. Once heated, season chicken and place, skin-side down, in the slow cooker. Cook, turning once, for 5-7 minutes or until browned. Transfer to a plate.
With the slow cooker still set to Browning, add celery, mushroom and shallot. Cook for 5 minutes or until golden and beginning to caramelise. Add garlic and cook for 1 minute. Pour in wine and cook for 3 minutes or until reduced by half. Add flour and tomato paste. Cook, stirring, for 2 minutes. Gradually add the chicken stock, stirring, until smooth. Bring to the boil. Stir in the tarragon and season.
Change the slow cooker setting to Low. Return the chicken to the slow cooker along with the potato. Cover and cook for 3 hours. Add carrot, tucking them into the mixture, and cover. Cook for a further 2-3 hours, adding the beans in the last 30 minutes of cooking, until chicken is cooked and vegetables are tender. Serve topped with extra tarragon sprigs.
