Rinse and cut off any fat from the chicken breasts. Place the chicken breasts in the slower cooker and sprinkle them with the salt and pepper.
Dice the onions, peppers, and garlic and spread on top of the chicken.
Add the barbeque sauce, starting with enough sauce to cover the chicken well. Stir to mix well.
Cover the slow cooker and turn it on. Cook 4 hours on high or 8 hours on low.
Remove the chicken from the slower cooker and shred or pull the meat apart with two forks, leaving the pieces medium sized.
Remove the barbecue broth and vegetables and set aside.
Place a layer of shredded chicken, 1/2 cup of the reserved broth, then some of the vegetables and a little more of the Sweet Baby Ray's Barbecue Sauce. Do this in layers until all the chicken and the vegetables are back in the slower cooker and about half of the broth is used. Very gently stir to combine, being careful not to mash the chicken.
Keep the slow cooker on warm or low until ready to serve.