Heat two tablespoons cooking oil in stock pot over medium-high heat; add chicken, cooking until browned (~10 minutes); remove chicken; set aside.
Put sausage in pot, cook until browned (~5 minutes); remove sausage; set aside.
Put shrimp in pot, cook until pink (~3 minutes); remove shrimp; set aside.
Reduce heat under stock pot to medium; melt butter; add two tablespoons cooking oil; whisk in flour, stirring continuously, scraping up any browned bits; continue to stir until roux is a chocolate milk color. (~15 minutes)
Return heat to medium-high; add bell pepper, onion and celery; stir to coat with roux; cook and stir until onions are translucent; add garlic, parsley, thyme, Cajun seasoning, cayenne pepper, bay leaves, salt and pepper; cook ~3 minutes more, until fragrant.
Return chicken, sausage and shrimp to stock pot; add chicken broth; stir well; bring to a bubbly boil; reduce heat to low; simmer uncovered 45 minutes.
Make rice; serve as pictured below; garnish gumbo with thinly sliced scallions (green onions), if desired.