Slice demis through the middle lengthwise to open them up like steak rolls. Cut each in half crosswise to make 4 pieces.
Cut chicken into 1 1/2-inch cubes, set aside. 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.
Measure out 1/3 cup of this marinade and reserve in a sealed container in the refrigerator.
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.
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.
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.