For dressing: In a small bowl, stir together mayonnaise, lemon juice and basil. Set aside.
For salad: In a medium bowl, combine chicken, Parmesan cheese, green onions and celery. Pour dressing over chicken mixture; toss to coat. Season to taste with salt and ground black pepper. Serve immediately on toasted wheat bread. Or to serve later, divide chicken mixture and pack in individual containers.
To tote: Pack chick mixture and bread separately in an insulated container with ice packs. Assemble to serve.