Preheat oven to 350 degrees F (175 degrees C). Spray an 8x8-inch baking dish with cooking spray.
Place the chicken breasts into the prepared baking dish.
Stir together the onion, ketchup, cider vinegar, brown sugar, ground mustard, ginger, and salt and pepper in a saucepan until the mixture is well combined. Bring to a boil over medium heat, and simmer until slightly thickened, about 15 minutes, stirring frequently.
Pour the sauce over the chicken, cover the dish with foil, and bake for 20 minutes.
Remove the foil, and bake the chicken breasts until no longer pink in the center and the juices run clear, and the sauce is slightly browned and bubbling, about 10 more minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
