Preheat the oven to 375 degrees F (190 degrees C). Lightly grease a baking dish.

Pierce sweet potato several times with a fork, then cut in half. Place in a microwave-safe dish; pour in about 1/2 inch of water. Microwave on high for 10 minutes.

Meanwhile, chop carrots, apple, and onion; arrange in the bottom of the prepared baking dish.

Brush barbecue sauce on both sides of the chicken breasts and lay atop the vegetables.

Remove sweet potato from the microwave. Peel and discard skin. Cut sweet potato into large chunks; scatter into the baking dish. Sprinkle cranberries over top.

Bake in the preheated oven until chicken is tender and no longer pink in the center, 45 to 55 minutes. Remove from the oven and allow to cool for 5 minutes before serving.