Heat olive oil in a large cast iron pot over medium-high heat. Add the sausage and minced garlic. Sauté about 5 minutes.
Stir in the onion, bell pepper, and celery and cook for 5 minutes until vegetables are softened.
Add the salt, cajun seasoning, cayenne, and cook for 2 minutes.
After adding the crushed tomatoes and chicken broth reduce heat to medium and cook for 5 minutes, add the sausage and butter. Turn the heat to medium-low and simmer for 20 minutes. During the last 2 minutes add Gumbo filé.
Add the shrimp and cook for 3 minutes.