In Dutch oven, combine chicken, celery, carrots, onions, pesto sauce mix and broth; mix well. Bring to a boil. Reduce heat to low; cover and cook 10 minutes.
Meanwhile, in medium bowl, combine flour, cornmeal and baking powder; mix well. Add milk, oil and egg; stir just until dry ingredients are moistened.
Drop dumpling dough by rounded tablespoonfuls onto boiling soup. Cover; cook 12 to 15 minutes or until dumplings are fluffy and no longer doughy on bottom.