Position a rack in the center of the oven and preheat to 350°F. On a foil-lined baking sheet, spread the sausage in an even layer and bake until the edges are browned, 35–40 minutes. Set aside and turn off the oven.
Meanwhile, in a shallow dish or pie plate, whisk together 2 cups of the flour and 1 tablespoon of the Cajun seasoning. Sprinkle the chicken all over with the remaining Cajun seasoning. Dredge each piece in the seasoned flour and transfer to a plate beside the stove.
Into a large pot or Dutch oven fitted with a deep-fry thermometer, pour the peanut oil to a depth of ½ inch and turn the heat to medium-high. When the temperature reads 350°F, working in batches, fry the chicken, turning once, until browned and crisp, about 10 minutes total. Transfer to a paper-towel-lined baking sheet. Position a cheesecloth-lined fine-mesh strainer over a heatproof measuring cup, then carefully strain the peanut oil. Wipe out the pot and return it to the stove.
Make the roux: Into the pot, pour ¾ cup plus 2 tablespoons of the reserved peanut oil (save the rest for another use) and turn the heat to medium-high. When it’s hot and shimmering, gradually whisk in the remaining flour and cook, whisking continuously, until the roux is butterscotch-colored, 2–3 minutes. Turn the heat to medium and continue to cook, whisking continuously, until the roux is reddish brown, 15–20 minutes more. Scrape the roux into a heatproof bowl, wipe out the pot, and return it to the stove.
To the empty pot, add the olive oil, 3 cups of the onion, 2 cups of the celery, 1½ cups of the bell pepper, and the bay leaves. Turn the heat to medium-high and cook, stirring occasionally, until the vegetables are lightly browned, 10–12 minutes. Turn the heat to medium and add the remaining onion, celery, and bell pepper and cook, stirring occasionally, for 3 minutes more. Turn the heat to low and add the filé, salt, cayenne, black pepper, thyme, white pepper, and garlic and cook, stirring continuously, until the filé dissolves into a smooth paste, 3–4 minutes. Gradually whisk in the chicken stock and bring to a boil. Add the reserved sausage, then turn the heat to low and cook, stirring occasionally and skimming any fat that rises to the surface, until the vegetables are soft and the liquid has reduced slightly, about 1 hour.
Add the reserved chicken and cook, stirring occasionally, until the meat is fully cooked, 30–35 minutes. Use tongs to transfer the chicken pieces to a cutting board, and when cool enough to handle, discard the bones and skin and coarsely chop the meat; set aside.
Bring the liquid to a boil and skim any fat that rises to the surface. Whisk 2 tablespoons of the roux into the boiling broth. Repeat, adding the roux spoonful by spoonful, until none remains. Turn the heat to low and cook, skimming any fat that rises to the surface, until the gumbo thickens slightly, about 25 minutes. Return the chopped chicken to the pot and turn the heat to medium. Cook, stirring gently, until the chicken is heated through, about 5 minutes. Serve with rice.