For the roast potatoes, preheat the oven to 220C/200C Fan/Gas 7. Put the potatoes into a saucepan and cover with cold water. Add a generous pinch of salt and bring to the boil. Simmer for 5 minutes, then drain and leave to dry out for a minute.
Put the duck fat (or oil) in a large roasting tin and heat in the oven until smoking. Add the potatoes and shake the tin so that the potatoes get well coated with the fat or oil. Season with plenty of salt and pepper.
Crush the garlic cloves, leaving their skins on, then add to the tin along with the rosemary. Roast in the oven for around 30 minutes, checking and turning over every so often until the potatoes are crisp and brown around the edges. The garlic may cook through sooner, in which case, remove it and set aside until the potatoes are cooked.
To make the sauce, put the sugar and 50ml/2fl oz of water in a saucepan and cook on a low heat, stirring regularly, until the sugar has dissolved. Add the kumquats and simmer for around 5 minutes until the kumquats are tender. Remove the kumquats from the syrup and set aside.
Turn up the heat and boil the syrup until it turns the colour of caramel. Add the red wine, red wine vinegar, shallots and star anise and return to the boil. Simmer until the volume of the liquid has reduced by half, then add the orange juice and chicken stock.
Again, simmer until the volume of the sauce has reduced by half. The sauce should be rich and syrupy. Taste for seasoning and add plenty of black pepper and salt if necessary. Return the kumquats to the sauce and set aside.
For the duck, score a criss-cross pattern through the skin and fat of the duck breasts, stopping just short of the flesh. Sprinkle with salt and pepper. Heat the oil (or duck fat) in a large oven-proof frying pan. When close to smoking point, add the duck breasts, skin-side down and fry until the skin is a rich golden brown and a great deal of fat has rendered out – around 6–7 minutes. Flip over and sear for a further 4 minutes. Remove from the pan and leave to rest.
Strain off most of the fat from the frying pan, then pour it in the sauce. Deglaze the pan and add back any juices which may have collected under the strained off fat. Finally add any juices from the resting duck. Simmer until piping hot and very syrupy, then serve the sauce over the sliced duck breasts.