Heat a large Dutch oven over medium heat. Add the sliced andouille sausage and cook until browned on both sides. Remove the sausage from the pot and set aside, leaving the drippings in the bottom of the pot.
Add the butter to the same pot and allow it to melt. Sprinkle in the all-purpose flour and begin stirring continuously with a wooden spoon or whisk.
Continue cooking and stirring as the roux progresses from pale to golden brown, then to the color of peanut butter, and finally to a deep, dark brown. Depending on your heat level, this can take 40–60 minutes. Don't walk away from the pot, especially as the roux gets darker, as it can burn quickly. (See pictures in post for examples).
Stir the diced onion into the dark roux and cook briefly. Add the celery and bell pepper and continue cooking until the vegetables begin to soften.
Stir in the minced garlic, Cajun seasoning, and dried thyme. Cook briefly, just until the garlic becomes fragrant.
Slowly pour in the chicken broth, stirring as you go to incorporate the roux into the liquid. Add the browned andouille sausage back to the pot and stir to combine.
Bring the gumbo to a gentle simmer, reduce the heat, and allow it to simmer for about 20–25 minutes so the flavors can develop. Taste and season with salt, black pepper, or additional Cajun seasoning as needed.
Add the shrimp during the final 5–7 minutes of cooking. Cook just until the shrimp are pink and opaque. Avoid overcooking the shrimp so they remain tender.
Ladle the hot gumbo into bowls and serve over warm white rice. Garnish with sliced green onions, if desired.