Special equipment: an instant-read thermometer
Toast chiles in a small dry heavy skillet over moderate heat until slightly darker, turning once with tongs, about 40 seconds total. Transfer to a small heatproof bowl, add boiling water, and soak until softened, about 20 minutes.
With a slotted spoon, transfer chiles to a blender. Add 1 cup soaking liquid and garlic and blend until smooth.
Cook 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 sugar has melted to a deep golden caramel, about 8 minutes. Carefully add orange and lime juices (caramel with steam vigorously and harden). Cook over moderately low heat, stirring, until hardened caramel is dissolved, about 5 minutes. Remove from heat.
With a sharp paring knife, score skin, through fat, on each duck breast in a crosshatch pattern, making score marks about 1 inch apart. Pat dry and sprinkle with salt and pepper.
Put 3 breast halves skin sides down in a 12-inch heavy skillet and turn heat to moderate. As fat is rendered, pour it into a heatproof bowl and reserve for another use. Cook breasts until skin is well browned, about 10 minutes. Turn over with tongs and cook until meat is browned, about 3 minutes. Transfer to a plate and brown remaining 3 breast halves in same manner.
Return all breast halves to skillet, cover, and cook over moderate heat until thermometer inserted horizontally into center of a breast registers 135°F for medium-rare, about 6 minutes. Transfer duck to a carving board and let stand, uncovered, while you make sauce. (Duck will continue to cook as it stands.)
Pour off all but 2 tablespoons fat from skillet. Add chile purée and any duck juices from plate 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 more. Whisk in butter until incorporated, then whisk in salt to taste.
Slice duck breasts and serve with sauce.
Cook's note: The USDA recommends cooking duck breasts to an internal temperature of 170°F to ensure that any harmful bacteria are killed, but since we prefer the meat medium-rare, we cook it to only 135°F. To our taste, that yields the perfect degree of doneness.