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.
Some family members enjoy a whole stuffed pepper in place of the half inside the chicken.
We normally cut the string as soon as the bacon shrinks to the chicken. Make sure you continue to baste with the barbecue sauce.
If you like to cook them in the oven, bake at 375 degrees F (190 degrees C) for 30 minutes or until chicken is done. Baste a couple times with the barbecue sauce and finish under the broiler to set the barbeque sauce.
Another option is to bake them at 400 degrees F (200 degrees C) without barbeque sauce for the first 20 minutes, turning once to help cook the bacon. Then reduce heat to 375 degrees F (190 degrees C), change pans to lose some of the bacon fat, cover with barbeque sauce, and and finish baking/broiling.
Per Serving: 323 calories;21.1 g fat;
5.3 g carbohydrates;
26.4 g protein;
88 mg cholesterol;
932 mg sodium.