Step 1

Adjust oven rack to middle position and heat oven to 425 degrees. Place flour in 12-inch skillet and bake, stirring occasionally, until color of ground cinnamon, 40 to 55 minutes. (As flour approaches desired color, it will take on very nutty aroma that will smell faintly of burnt popcorn, and it will need to be stirred more frequently.) Transfer flour to medium bowl and let cool. (Toasted flour can be stored in airtight container in cool, dark place for up to 6 months.)

Step 2

Heat oil in Dutch oven over medium heat until shimmering. Add onion, bell pepper, and celery and cook, stirring frequently, until softened, 5 to 7 minutes. Stir in thyme, garlic, paprika, bay leaves, cayenne, 1/4 teaspoon salt, and 1/4 teaspoon pepper and cook until fragrant, about 1 minute. Stir in 2 cups broth. Add chicken in single layer (chicken will not be completely submerged in liquid) and bring to simmer. Reduce heat to medium-low, cover, and simmer until chicken is fork-tender, 15 to 17 minutes. Transfer chicken to plate.

Step 3

Slowly whisk remaining 2 cups broth in small increments into toasted flour until thick, smooth, batter-like paste forms. Increase heat to medium and slowly whisk paste into gumbo, making sure each addition is incorporated before adding next. Stir in andouille. Simmer, uncovered, until gumbo thickens slightly, 20 to 25 minutes.

Step 4

Once cool enough to handle, shred chicken into bite-size pieces. Stir chicken and scallions into gumbo. Remove pot from heat, stir in vinegar, and season with salt to taste. Discard bay leaves. Serve, passing hot sauce separately. (Gumbo can be refrigerated in airtight container for up to 3 days.)