Heat olive oil and butter in a large saucepan over medium heat. Add the onion and andouille sausage and cook and stir until the onion starts to brown, about 10 minutes. Stir in garlic and cook until fragrant, 1 to 2 minutes.
Mix in crushed tomatoes, green bell peppers, zucchinis, Cajun seasoning, hot sauce, and chicken broth; bring mixture to a boil, reduce to a simmer, and cook uncovered until the liquid cooks off and the mixture is thick, about 15 minutes. Stir in chicken and shrimp and simmer until heated through, 1 to 2 minutes.
For best flavor, use fresh gulf shrimp and andouille, but frozen shrimp or pre-cooked shrimp from your supermarket’s seafood counter also work well in this recipe. If you can’t find andouille sausage, a flavorful smoked pork sausage is a good substitute.
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