Heat the oven to 170 degrees C. Place the duck breasts in a dry oven dish, skin side down and fry until the skin is crisp. Bake them in the oven for 12 minutes or until they reach an internal temperature of 63 degrees C. Leave to rest for 10 minutes. Boil two carrots in the white wine vinegar, sugar, water and apple juice until they are tender. Add the butter and blend with a hand blender. Grill the remaining carrots in a grill pan until they are tender. Brush the duck breasts with hoisin sauce and serve.