The day before, put the figs in a bowl with the port. Cover with clingfilm (plastic wrap) and put in the fridge to marinate overnight. The next day, preheat the oven to 170°C (325°F/Gas 3).
Season the duck legs with salt and pepper, then dip them in the flour. Shake of any excess flour. Put the oil in a large lidded flameproof and ovenproof casserole dish (Dutch oven) over a medium heat. When the oil is hot, add the duck legs and cook for 5 minutes on each side until golden. Remove with a slotted spoon to a plate and set aside.
Pour off the excess oil from the casserole dish (Dutch oven) and reduce the heat. Add the butter, mint stalks, sugar and balsamic vinegar. Cook over a low heat for 4–5 minutes until lightly caramelized.
Return the duck legs to the casserole dish (Dutch oven). Remove the figs from the port with a slotted spoon and pour the port over the duck legs. Add the chicken stock and bring to the boil. Reduce the heat to a simmer and tuck in the marinated figs. Cover with a lid and transfer to the preheated oven. Braise for 2–2½ hours, or until the duck is tender. Add a little more stock if it starts to dry out.
Carefully remove the duck legs and figs with a slotted spoon and divide between the serving plates. Strain the cooking sauce through a fine sieve (strainer) into a small pan and bring to the boil. Add the chopped mint leaves, pour over the duck legs and serve immediately.