Preheat the oven to 425°F. Place the chicken breasts on a cutting board and slice carefully into the thickest part of the side of each breast, creating a slit that is about 4 inches across, without slicing through the entire breast. Pat each breast dry with a paper towel and then sprinkle with salt and pepper.

Place the broccoli and the corn in a medium size glass bowl, along with 2 tablespoons of water. Cook in the microwave for about 3 minutes until the broccoli is bright green and slightly tender. Alternatively, you can steam the broccoli and corn to soften it. Add the cheese to the bowl with the broccoli and stir to combine.

Use a spoon to fill each chicken breast with the cheesy vegetable mixture and pin each one closed with a toothpick. Set the stuffed breasts in a 9x13 baking dish and pour barbecue sauce over them.

Bake for 15 to 25 minutes, until the chicken reaches 165°F. Remove from the oven and let rest for 5 minutes before removing the toothpicks and slicing or serving whole. Serve with additional barbecue sauce, if desired.