Pat the duck skin dry with some paper towel then score with a sharp knife and season with a little salt.
Place the duck, skin side down, into a large fry pan, over a medium heat to begin rendering the skin. Season the flesh side with a little more salt and cook like this for 8-10 minutes, adjusting the heat higher if needed so the skin is caramelising and the fat rendering out. Flip the breast and continue to cook for 4 more minutes then remove from the pan to rest.
Remove half the duck fat from the pan, reserving.
Add the onion and garlic to the pan and cook over a medium heat for 3 minutes to soften then add the potatoes, spices, half the curry leaves and cook for 2 minutes more.
Add the tomato, 200ml water and ¾ of the coconut milk, season lightly, bring to a simmer, cover, and cook until the potatoes are tender, approx. 12 minutes.
Heat the remaining duck fat in a small saucepan and add the curry leaves, cooking quickly until they become crisp and bright. Remove to a paper towel lined plate.
Thinly slice the duck breast and add to the pan with any juices, the chickpeas, broccolini and stir gently. Cover and simmer for a further 6-8 minutes then scatter with the chopped coriander, crisp curry leaves, drizzle with remaining coconut milk and serve with steamed rice.