Make the Roux: In a large, heavy-bottomed pot, melt the unsalted butter over medium heat. Gradually whisk in the flour, stirring constantly to prevent burning. Cook the roux, stirring continuously, until it turns a dark brown color, about 10-15 minutes.
Add the Holy Trinity: Add the chopped onion, bell pepper, and celery to the pot. Cook, stirring frequently, until the vegetables are soft and the onions are translucent, about 10 minutes. Add the minced garlic and cook for another 2 minutes.
Simmer the Sauce: Stir in the chicken broth, diced tomatoes, tomato paste, bay leaves, Cajun seasoning, smoked paprika, cayenne pepper, dried thyme, salt, and black pepper. Bring to a simmer and cook for 20 minutes, stirring occasionally, until the sauce thickens and the flavors meld together.
Add the Shrimp: Add the shrimp to the pot and cook until they are pink and opaque, about 5 minutes. Stir in the chopped parsley and green onions.