Heat oil in a 3-quart Dutch Oven over medium-high heat until it begins to smoke.
Add flour, whisking constantly, and cook for 1 minute.
Reduce heat to medium and cook, whisking constantly, until roux is the color of milk chocolate, about 12 - 15 minutes.
Add celery, onions, and peppers. Cook, stirring constantly, until soft, about 5 minutes.
Stir in salt, cayenne, black and white peppers, basil and thyme. Cook until fragrant, about 1 minute more.
Add 2 cups chicken stock. Bring to a boil. Cook until thickened, about 5 minutes.
Meanwhile, heat butter in a 12-inch skillet over medium-high heat.
Add chicken. Cook, turning once, until lightly browned, 4 - 6 minutes.
Transfer chicken and butter to Dutch oven.
Pour remaining chicken stock into skillet. Stir to scrape up any browned bits and then pour into Dutch oven along with andouille. Cook, stirring occasionally, until thick and chicken is cooked through, about 10 minutes more.
Remove pan from heat. Stir in scallions, reserving a few to sprinkle over the top, and serve étouffée with prepared rice.