In 5-quart Dutch oven or stockpot, melt butter over medium heat. Add carrots, celery and onion; cook about 5 minutes, stirring frequently, until carrots are crisp-tender.
Stir in broth, milk, uncooked pasta and sauce mix (from Tuna Helper box), chicken and bay leaf. Heat to boiling. Reduce heat; simmer uncovered 12 to 15 minutes, stirring occasionally, until pasta and vegetables are tender.
Remove bay leaf from soup; ladle soup into bowls. Place pesto in small resealable food-storage plastic bag; seal bag. Cut off small corner of bag; drizzle pesto over soup.