Bring 3.5 quarts of water to a rolling boil in a large pot. Add the Cajun seafood boil spice, celery, quartered onion, green bell pepper, smashed garlic cloves, bay leaf, lemon quarters, and thyme sprigs. Bring back to a boil, then reduce heat to medium and simmer for 15 minutes to allow the aromatics to fully infuse the broth and develop deep, authentic Louisiana flavor. I like using smashed garlic cloves at this stage rather than minced—they release their oils more effectively into the liquid.
Add the sliced andouille sausage and sweet potato cubes to the simmering broth. The sausage will release its smoky, spiced oils into the liquid, further building the flavor base. Maintain a gentle simmer and cook for 15 to 20 minutes until the sweet potatoes are nearly tender but still hold their shape. Start preparing the butter sauce and remoulade while these ingredients cook (see Steps 3 and 4).
While the potatoes and sausage cook, melt 3/4 cup butter in a small saucepan over low heat. Add 2 minced garlic cloves, 3/4 teaspoon thyme, and 3/4 teaspoon Cajun spice. Stir gently until fragrant and the garlic softens without browning, about 2 to 3 minutes. Keep warm on low heat until serving.
In a small bowl, whisk together 1 cup mayonnaise, 2.5 tablespoons Creole mustard, 2 tablespoons ketchup, 1 tablespoon hot sauce, 4 minced garlic cloves, 2 teaspoons thyme, 1.5 teaspoons Worcestershire sauce, 1 teaspoon paprika, and 1/2 teaspoon cayenne until smooth and well combined. Taste and adjust seasonings if needed. Set aside at room temperature until serving. I love making this tangy sauce ahead since it actually tastes better as the flavors marry while it sits.
When the sweet potatoes are nearly tender, add the corn halves to the boil and cook for 5 minutes. Remove the pot from heat and immediately stir in the shrimp, distributing them evenly throughout the broth. Cover the pot and let it sit undisturbed for 4 to 5 minutes—residual heat will gently poach the shrimp to perfect tenderness without overcooking. The shrimp will turn pink and opaque when done.
Using a large colander, carefully drain the shrimp boil mixture, reserving some of the flavorful broth if desired for serving alongside. Transfer the drained shrimp, sausage, corn, and sweet potatoes to a large platter or serving bowl. Drizzle generously with the warm garlic butter sauce from Step 3 and serve with the remoulade sauce from Step 4 on the side. Provide plenty of napkins—this is authentic Louisiana comfort food at its finest!