Arrange chicken pieces and potatoes on the baking sheets. Sprinkle with salt and pepper. Bake chicken and potatoes for 20 minutes.
Meanwhile, prepare bourbon sauce. Combine the sauce ingredients in a medium saucepan over medium heat. Whisk until the ingredients are incorporated. Continue to cook on low until the mixture begins to thicken.
After the chicken has been baking for 20 minutes, add the broccoli to the same pan. Baste chicken pieces with the sauce and continue to bake chicken for another 18-20 minutes or into an internal temperature registers 165 degrees.