In a heavy 6–8 quart Dutch oven over medium-low heat, whisk together the vegetable oil and all-purpose flour until smooth, then cook, stirring constantly with a flat wooden spoon, until the roux turns the color of dark chocolate, 30–45 minutes; adjust heat to keep it deeply browned but not burnt.
Stir in the yellow onion, celery, and green bell pepper; cook until softened and glossy, 5–7 minutes. Add the garlic and cook 1 minute until fragrant.
Gradually whisk in the chicken stock until smooth. Add the bay leaves, dried thyme, black pepper, cayenne pepper, and kosher salt. Bring to a gentle boil, then reduce to a simmer.
Add the andouille sausage and the chicken thighs. Simmer uncovered, stirring occasionally and skimming any fat that rises, until the chicken is tender and the liquid lightly coats a spoon, 45–60 minutes.
Turn off the heat and stir in the scallions and flat-leaf parsley. Let the pot rest 10 minutes to settle.