In a large mixing bowl, combine the shredded chicken, mayonnaise, olive oil, chopped celery, cucumber, chopped red bell pepper, Dijon mustard, and balsamic. Mix well until all ingredients are evenly incorporated.
Season with salt and pepper to taste. Adjust the amount of mayonnaise and mustard if necessary.
Lay out 8 slices of whole grain bread.
Place a leaf of lettuce on 4 of the slices.
Add a couple of tomato slices on top of the lettuce.
Generously spread the chicken salad mixture over the tomato slices.
Top with the remaining slices of bread to complete the sandwiches.
Cut each sandwich in half, if desired, and serve immediately.