Cut the pummelo in half. Juice and zest one half. Peel the other half of the pummelo with a sharp knife, cutting away as much of the peel and bitter white pith as you can. Separate it into sections, cutting them in half if the pummelo is very large. If the juice does not make about 2/3 cup, you can add water or a little grapefruit juice to make up the rest.
Preheat the oven to 350°F.
Score the duck breasts with a cross hatch pattern.
Mix the honey, brown sugar, ginger, and Cayenne pepper in a small bowl. Use a little less than half to brush on the duck breasts
Heat the olive oil in a large ovenproof skillet. Once heated, add the duck breasts, skin side down, and sear for about 6 minutes over a medium-low setting. Turn and cook for another 2 minutes on the other side.
Place the skillet in the oven and roast for about 5 minutes. Transfer to a cutting board and tent with foil. Allow it to rest for at least 4 minutes.
Transfer the remaining honey mixture to a small saucepan, and add about 1 tablespoon of the duck fat from the skillet. Stir together, and cook over a medium setting until the mixture is a light golden brown.
Add the chicken stock, cream, basil, and remaining pummelo juice. Simmer until the sauce is reduced to about 1/2 cup.
Stir in the unsalted butter to melt, and then the pummelo sections. Season with salt and pepper to taste.
Serve the duck breasts with the sauce poured on top.