To make the roux, whisk together the flour and oil in a large, heavy-bottomed stock pot or Dutch oven.
Set the pan over medium heat and cook for 30 to 60 minutes, until the mixture is chocolate-brown, stirring almost constantly. Keep a close watch so the roux doesn’t burn.
Brown the sausage in a separate skillet over medium-high heat. Cook until browned on both sides. Transfer the sausage to a plate, leaving the grease in the pan.
Add ½ cup of the broth to the pan used to sear the sausage. Bring to a simmer, scraping up any browned bits from the bottom of the pan.
When the roux is ready (chocolate-brown), add the broth mixture and whisk to combine.
Add the remaining 5 ½ cups of broth, celery, bell pepper, onion, green onions, parsley, garlic, and 1 tablespoon of the Cajun seasoning.
Increase the heat to medium-high and bring to a simmer.
Cook for 5 to 7 minutes, until the vegetables are tender-crisp.
Add the chicken, sausage, and shrimp and simmer for 2 minutes, until heated through (or slightly longer if using raw shrimp).
Taste and add more Cajun seasoning if desired.
Ladle the gumbo into bowls, add a scoop of rice, and serve.