Use a sharp knife to score across the duck breasts 4 times through the skin and fat but just barely to the meat. Rub the skin with salt, cayenne, and black pepper.
Preheat an ovenproof skillet over medium-high heat. Lay the breasts in the skillet skin-side down and fry until the skin is brown and crisp, about 5 minutes. Use a spoon to carefully discard any excess fat from the bottom of the skillet. Turn the breasts over and cook for 1 minute.
Place the skillet into the preheated oven and roast until the internal temperature of the thickest part of the breasts reach 160 degrees F (71 degrees C) for well done, or the breasts reach desired doneness.
Remove the duck breasts from the skillet and cover with foil. Set aside to rest. Pour off excess fat from the skillet. Place the stock, honey, soy sauce, rice wine, ginger, tomato sauce, chili powder, and lime juice in the skillet. Whisk the sauce over high heat, bring to a boil and cook until the sauce thickens, about 2 minutes. Slice the duck breasts thinly, arrange on serving plates, and pour the sauce over the top.
