Lightly combine the canned chicken, eggs, Parmesan cheese, pine nuts, pickle relish, chopped onion, mayonnaise, cole slaw dressing, and salt and pepper to taste in a large bowl. Cover and refrigerate for 30 minutes to blend the flavors.
To assemble sandwiches, spread a generous serving on each of 5 slices of sandwich bread, and place lettuce leaves, sliced tomato, sliced avocado, and Monterey Jack cheese slices over the salad. Top with remaining sandwich bread slices.
