In a large Dutch oven, heat butter over medium heat. Add andouille; cook, stirring frequently, until browned, 6 to 8 minutes. Using a slotted spoon, transfer andouille to a medium bowl.
Add shrimp to pot; sprinkle with ½ teaspoon Creole seasoning. Cook, stirring frequently, until pink and firm, 2 to 3 minutes. Transfer shrimp to bowl with andouille.
Add onion, bell pepper, celery, and garlic to pot; cook until vegetables are softened, 3 to 4 minutes. Add tomatoes, broth, parsley, thyme, brown sugar, salt, and remaining 1 tablespoon Creole seasoning. Break up tomatoes using a wooden spoon; bring to a boil. Reduce heat to medium; simmer until slightly thickened, about 15 minutes.
Stir in andouille mixture and cooked pasta, tossing to coat. Garnish with parsley and thyme, if desired.
Recipe by Louisiana Cookin' at https://louisianacookin.com/shrimp-andouille-pastalaya/