Heat a medium saute pan over medium-high heat and add the oil.
Cut a crosshatch pattern into the skin of the duck breasts. Be careful to not cut through to the flesh.
Season the skin side with the curry and the flesh side with salt. Place in the pan, skin side down. Cook on the skin side to render of some of the fat and crisp up the skin, 5 to 7 minutes. Once the skin is crisp, flip the breasts, reduce the heat to medium, and cook on the flesh side for about 4 minutes for medium-rare. Once cooked, remove from the pan and allow the meat to rest for 5 minutes.
Make the sauce in the same pan used for the duck. (Wipe out the pan if you have burned bits on the bottom, and add some butter.) Place the pan over medium-high heat and add 1 tablespoon butter. Deglaze the pan with the pomegranate juice. Allow the juice to simmer and reduce by half. Turn off the heat and stir in the remaining 2 tablespoons butter. Taste and season with a pinch of sugar if the sauce tastes too tart.
When plating, slice the duck breast thinly and spoon the warm sauce over the duck. Serve with Mashed Sweet Potatoes with Goat Cheese and Sauteed Julienned Vegetables.
Peel and slice the sweet potatoes into 1/2-inch pieces. Place the potatoes in a large saucepan, fill with water to cover the potatoes, and add some salt. Bring the water to a boil over medium-high heat. Cook the potatoes until fork tender, about 13 minutes. Stick a fork in a potato piece and if it goes in easily, it's done.
Drain the potatoes well. Transfer them to a food processor and blend until smooth. Add the goat cheese, honey, ginger, cinnamon, butter, and 1/2 teaspoon salt. Blend again until everything is well combined. Transfer to a bowl and let cool for 1 minute.
Spoon the potatoes into a zip-top plastic bag or a pastry bag with a tip for piping, or spoon directly onto plates. If piping, cut a hole in the tip and pipe onto the plate.
Heat a saute pan over medium heat and add some oil. Saute the julienned vegetables and season with salt and pepper. Cook until the vegetables are just softened, 3 to 5 minutes. Taste a bite and cook for another minute if the vegetables are still hard.