Melt butter in a large Dutch oven over medium-low heat. Sprinkle flour over top; cook, stirring constantly, until dark caramel colored, about 10 minutes. Be careful not to let the roux burn.
Add onion, bell pepper, and celery, increase the heat to medium or medium-high, and stir until softened, about 5 minutes. Stir in garlic; cook for 1 more minute.
Whisk in the chicken broth. Increase the heat to high and whisk until smooth.
Add the diced tomatoes (with their juices), whisk, and bring to a boil.
Reduce heat to low and stir in the thyme, oregano, bay leaf, paprika, and cayenne pepper. Simmer for 30 minutes, stirring occasionally.
Add the shredded chicken, ham, oysters, and okra to the pot. Return to a simmer and cook for 10 more minutes. Remove bay leaf.
Remove the pot from the heat and stir in the green onions, parsley, and filé powder. Taste and season with salt and more cayenne, if necessary. Serve with warm rice.