Stirring constantly (Bienvenu uses a wooden spoon), cook until the roux is medium-dark-brown (the color of dark peanut butter, although some cooks prefer the color of chocolate). This will take about 20 to 25 minutes.
Carefully add the onions, celery and green peppers to the hot cooked roux. You may want to stand back a little as you add them (or take the pan off the heat) as wet vegetables can make the mixture steam, pop and splatter (and if you should get splattered, you will understand why they call it "Cajun napalm"). Then cook, stirring, until the vegetables are soft, about 8 more minutes.
Warm the chicken broth and, if you started cooking the roux in a skillet, you will want to transfer everything to a Dutch oven; stir to mix.
Add the chicken pieces and thyme, and bring to a boil. Reduce heat and simmer, partially covered, 11/2 hours, stirring occasionally and skimming foam off the top from time to time if necessary.
Add the andouille sausage and cook for another hour, or until the chicken is tender. Before serving, use a spoon to skim off any excess fat or oil.
To serve: Put a big spoonful of steamed rice into deep soup bowls. Ladle the gumbo over the rice. Sprinkle each bowl of gumbo with green onions and parsley. If desired, pass the filé powder at the table for guests to add their own to taste.