Adjust oven rack to middle position and heat oven to 425 degrees. Place flour in Dutch oven and stir, stirring occasionally, until color of ground cinnamon, about 40 to 55 minutes. Remove pot from oven. Transfer flour to medium bowl and let cool. Wipe pot clean with paper towels.

Heat oil in now-empty pot over medium heat shimmering. Add onion, bell pepper, and celery and cook, stirring frequently, until softened, 7 minutes. Stir in thyme, garlic, paprika, bay leaves, cayenne, ½ teaspoon salt, and ¼ teaspoon pepper. Cook until fragrant, about 1 minute. Stir in toasted flour until completely incorporated. Whisk in broth in single layer (chicken will not be completely submerged in liquid) and bring to simmer. Reduce heat to medium-low and cook until chicken is fork-tender, 15 to 17 minutes. Transfer chicken to cutting board, let cool slightly, then shred into bite-size pieces using 2 forks.

Meanwhile, slowly whisk remaining 2 cups broth 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.

Stir chicken and scallions into gumbo. Off heat, stir in vinegar and season with salt to taste. Discard bay leaves. Serve, passing hot sauce separately.