In a medium Dutch oven, heat butter over medium heat; whisk in flour. Cook, stirring constantly, until a dark brown roux forms, about 6 minutes.
Add sweet onion, bell pepper, celery, green onion, and garlic; cook until fragrant and onion begins to soften, about 3 minutes. Add tomatoes, tomato paste, Worcestershire, bay leaves, and thyme. Slowly add broth, stirring constantly, and bring to a boil. Stir in salt and pepper. Reduce heat to low, and simmer, stirring frequently, for 30 minutes.
Stir in andouille; increase heat to medium, and cook, stirring frequently, until andouille is heated through, about 10 minutes. Add additional broth, if desired. Discard bay leaves. Serve over hot cooked rice. Garnish with green onion, if desired.
Recipe by Louisiana Cookin' at https://louisianacookin.com/andouille-and-bell-pepper-sauce-piquante/