Make the roux: In a large heavy-bottomed pot or Dutch oven, heat the oil over medium heat. Gradually whisk in the flour, a little at a time, until a smooth paste forms.
Continue cooking the roux, stirring constantly, until it turns a deep reddish-brown color, about 20-30 minutes. Be careful not to burn the roux.
Add the onion, green bell pepper, and celery to the roux and cook, stirring constantly, until the vegetables are softened, about 5 minutes.
Stir in the garlic, chicken broth, diced tomatoes, bay leaf, thyme, cayenne pepper, salt, and black pepper.
Bring to a boil, then reduce heat to low, cover, and simmer for 1 hour, stirring occasionally.
Stir in the shredded chicken and andouille sausage and cook for 15 minutes.
Stir in the shrimp and green onions and cook for 5 minutes, or until the shrimp are pink and cooked through.
Remove the bay leaf.
Serve the gumbo hot over cooked white rice.