Melt butter in a large pot over medium heat. Add sausage and cook for 2 minutes. Add celery, onion, bell pepper, and garlic; cook until soft, 5 to 7 minutes. Add chicken and cook until no longer pink inside and juices run clear, 5 to 7 minutes more. Add tomatoes, paprika, Creole seasoning, and cayenne pepper; bring to a simmer and cook for 10 minutes.

Add broth to the pot and return to a simmer. Add penne and simmer for 8 minutes. Add shrimp. Continue to simmer, stirring constantly, until pasta is tender but firm to the bite and shrimp are cooked through, about 7 minutes more.

Stir in cream. Stir in thyme, oregano, and basil. Cook until just heated through, 3 to 5 minutes more. Serve topped with green onions and Parmesan cheese.