Step 1Toast the chiles in a small dry skillet over moderate heat until slightly darker, turning once with tongs, about 40 seconds total. Transfer to a small heatproof bowl, add the boiling water and soak until softened, about 20 minutes.
Step 2With a slotted spoon, transfer the chiles to a blender. Add 1 cup soaking liquid and the garlic and blend until smooth.
Step 3Cook the sugar in a dry 1 1/2 -quart heavy saucepan over moderate heat, undisturbed, until it begins to melt. Continue to cook, stirring occasionally with a fork, until the sugar has melted to a deep golden caramel, about 8 minutes. Carefully add orange and lime juices (caramel will steam vigorously and harden). Cook over moderately low heat, stirring, until hardened caramel is dissolved, about 5 minutes. Remove from heat.
Step 4With a sharp paring knife, score the skin, through the fat, on each duck breast in a crosshatch pattern, making score marks about 1 inch apart. Pat dry and sprinkle with salt and pepper.
Step 5Put three breast halves skin sides down in a 12-inch heavy skillet and turn heat to moderate. Cook until skin is well browned, about 10 minutes. Turn over with tongs and cook until meat is browned, about 3 minutes.
Step 6Return all breast halves to skillet, cover and cook over moderate heat until a thermometer inserted horizontally into center of a breast registers 135 degrees for medium-rare, about 6 minutes. Transfer the duck to a carving board and let stand, uncovered, while you make the sauce. (The duck will continue to cook as it stands.)
Step 7Add the chile puree and any duck juices from the plate to the skillet and cook over moderately high heat, stirring and scraping up any brown bits, until thickened, about 6 minutes. Add caramel and any juices accumulated on carving board and simmer for 5 minutes. Whisk in butter until incorporated, then whisk in salt to taste.
Step 8Slice the duck breasts and serve with the sauce.