In a large 6-quart pot over medium-low heat, melt butter. Once melted, add flour and whisk together until smooth. Continue cooking, whisking constantly until the roux is a deep brown color (about 30-40 minutes). Be careful not to let it burn.
Remove from heat. Continue whisking until the mixture is cooled slightly. Set aside.
While the roux is cooling, add the chopped celery, onion, pepper, and garlic to the bowl of a food processor. Process until finely chopped.
Add the vegetable mixture to the roux and stir to combine. Place the pot over medium-low heat and continue to heat until vegetables are tender (8-12 minutes). Slowly whisk in broth and heat to boiling (15-20 minutes).
Reduce heat to low. Add sausage, stewed tomatoes, tomato sauce, hot sauce, sugar, Cajun seasoning, and thyme. Stir to combine. Add bay leaves and simmer, uncovered for 45 minutes.
After simmering for 45 minutes, add 2 teaspoons of gumbo filé powder, stirring to combine. Continue to simmer for 15 more minutes.
Remove the bay leaves. Then, add shrimp and Worcestershire sauce. Continue to cook over low heat for 45-60 minutes to give the flavors ample time to meld.
Stir in the remaining gumbo filé powder.
Serve over cooked white rice.