1. Slice demis through the middle lengthwise to open them up like steak rolls. Cut each in half crosswise to make 4 pieces.
2. Cut chicken into 1 1/2-inch cubes. Put lemon juice and vinegar in medium bowl. Whisk in oil in a slow steady stream. When fully incorporated, whisk in bay leaves, garlic, Italian seasoning, salt, and pepper.
3. Measure out 1/3 cup of this marinade and reserve in a sealed container in the refrigerator.
4. Pour remaining marinade into 1-gallon freezer-weight ziplock bag. Add chicken, press out the air, seal, and refrigerate for 2 days. Chicken will appear white from the marinade; this is normal and helps to tenderize the meat.
5. Remove chicken and container of reserved marinade from cooler 20 minutes before serving. Heat grill to medium-high. Brush grill grate and coat with oil. Grill chicken pieces directly on the grill until browned in spots and no longer pink in the center, 8 to 10 minutes, turning often and drizzling with the reserved marinade from the container during first 5 to 7 minutes. Avoid overcooking; chicken should be very moist and register about 160º on an instant-read thermometer.
6. Open and lightly toast bread pieces on the side of the grill, and then put the chicken on the bread. Drizzle on reserved marinade and serve warm.
TIP: To enrich your sandwich, spread on a little mayonnaise before adding the chicken.
911 calories, 60 g protein, 46 g carbohydrate, 52 g fat, 134 mg cholesterol, 1892 mg sodium, 3 g dietary fiber