Heat oil in a large frying pan over medium-high heat; add chicken breasts in a single layer, skin side down, and cook until browned on that side, about 5 minutes. Remove with a slotted spoon and set aside. Discard oil in pan, leaving any browned bits on the bottom.
Add chiles, soy sauce, vinegar, sugar, and 1 1/2 cups water to pan; bring to a boil over high heat, then reduce heat and simmer until slightly thickened, about 5 minutes.
Return chicken, skin side up, to pan. Cover and simmer until no longer pink in the center, about 15 minutes. Transfer to a serving dish, cover with foil, and keep warm in a 200° oven. Meanwhile, simmer the sauce until it is reduced and coats the back of a spoon, about 25 minutes longer.
Remove chiles and discard. Spoon sauce over chicken and garnish with cilantro.
Dried chiles: Distinct in taste and texture from their fresh sources, dried chiles-—such as the ones used in this recipe-—offer complex and concentrated flavor, making them a great addition to sauces and stews.
How to buy: Look for whole, unbroken chiles of uniform color that are still flexible (dusty, brittle chiles have lost much of their flavor). Note: Fresh and dried versions of the same chiles often go by different names. For instance, an ancho is a type of dried poblano, chipotles are dried and smoked jalapeños, and pasillas are dried chilacas. To add to the confusion, fresh poblanos and anchos are sometimes called pasillas in the West.
How to use: Dried chiles are usually ground into a powder; plumped and puréed; or added whole or chopped to liquids and sauces. Rinse or wipe with a damp paper towel and remove stems before using.
How to store: Keep dried chiles in an airtight container in a dry, dark, and cool place. Use within six months, discarding any brittle or discolored chiles.
Note: Nutritional analysis is per serving.