In a large pot or Dutch oven, heat vegetable oil over medium heat.
Gradually whisk in the flour, stirring constantly to make a roux. Cook for about 20 minutes, stirring until it turns a dark brown color (like chocolate).
Add diced onion, bell pepper, celery, and garlic to the roux. Sauté for about 5-7 minutes, until the vegetables are soft.
Stir in the sliced sausage and cook for another 5 minutes until browned.
Pour in the chicken broth and add diced tomatoes, bay leaves, Cajun seasoning, and thyme. Bring to a boil, then reduce heat to simmer.
Allow the gumbo to simmer for about 20 minutes, stirring occasionally.
Add the shrimp and continue to cook for an additional 5-7 minutes, or until the shrimp are cooked through and pink.
Season with salt and pepper to taste, then remove bay leaves before serving.
Serve hot over cooked white rice and garnish with chopped green onions.