Preheat a gas grill for medium heat (350 degrees F (175 degrees C)), or arrange charcoal briquettes on one side of the barbeque. Lightly oil the grate.
Slice chicken breasts in half lengthwise. Transfer to a resealable plastic bag, seal, and pound to 1/4-inch thickness with a meat mallet. Season with salt and pepper.
Mix Monterey Jack cheese and cream cheese together in a bowl. Fill jalapeno halves with cheese mixture, using about 1 tablespoon in each one.
Place 1 stuffed jalapeno with the cheese side down in the middle of each chicken breast half; fold into a pouch. Wrap each chicken breast completely with three pieces of bacon. Tie 4 ways with roasting string.
Cook on the preheated grill over indirect heat, turning every 5 minutes and basting with barbecue sauce, until juices run clear, about 30 minutes.