In a large Dutch oven, combine chicken broth, water, bay leaf, onion powder and thyme. Bring to a boil over medium heat. Reduce heat and let simmer while making dumplings.
In a large bowl, blend together flour, Parmesan cheese and salt. Cut in shortening with a pastry blender until mixture resembles coarse crumbs. Stir in chopped basil.
Add milk and stir until dough forms a ball. Divide dough in half, and place half on a lightly floured surface. Roll out dough to a 1/8-inch thickness. Cut dough into strips, approximately 1 x 3 inches in size. Repeat with remaining half of dough.
Bring chicken broth to a full rolling boil; add butter. Drop dumplings, a few at a time, into boiling broth, letting broth return to a boil before adding more. After all the dumplings have been added, cook for 15 minutes.
While dumplings are cooking, pull meat from rotisserie chicken, discarding skin and bones. Shred or chop chicken. (You should have approximately 3 cups of meat.) Add chiclken to dumplings. Season with pepper to taste.
Reduce heat to a simmer and continue to cook for 5 to 10 minutes, or until chicken is heated through. Garnish with additional fresh basil, if desired.