Combine the vegetable oil and flour in a small saucepan. Cook over medium heat, stirring constantly, until the mixture is a dark brown color and smells nutty, 20–25 minutes. Be careful not to let it burn. Set aside.
Prepare the grill for indirect medium heat, 375°–425°F. Place a large Dutch oven in the grill and preheat for 15 minutes. Add the butter to the pot to melt it, then add the bell pepper, celery, and onion. Close the grill lid and cook the vegetables until softened, 10–12 minutes, stirring occasionally.
Set the grill to 400°F. Place a large Dutch oven in the grill and preheat for 15 minutes. Add the butter to the pot to melt it, then add the bell pepper, celery, and onion. Close the grill lid and cook the vegetables until softened, 10–12 minutes, stirring occasionally.
Stir in the cayenne and filé powder and cook for about a minute.
Add the tomatoes, close the grill lid and cook until they start to break down, 10–12 minutes, stirring occasionally.
Add the turkey, andouille sausage, and stock to the Dutch oven.
Stir the prepared roux into the gumbo until fully incorporated. Close the grill lid and cook until slightly thickened and the flavors have melded, about 30 minutes, stirring occasionally.
Add the okra and continue to simmer, covered, until tender and the gumbo has thickened further, about 30 minutes.
Taste the gumbo, add the salt and pepper as needed before serving.
