How to Make It
Prepare the Ketchup: Cook butter and flour in a small saucepan over medium, whisking constantly, until smooth, about 3 minutes. Continue cooking, whisking often, until very mixture turns the color of milk chocolate, 10 to 12 minutes. Carefully pour in stock (mixture will boil rapidly); whisk in tomato paste until smooth, about 1 minute. Bring to a boil over high; reduce heat to medium, and simmer, stirring occasionally, until reduced by about half, 15 to 20 minutes. Remove from heat; stir in vinegar, hot sauce, and salt. Let cool to room temperature, about 10 minutes.
Meanwhile, prepare the Patties: Melt butter in a large skillet over medium-high. Add onion, celery, and bell pepper; cook, stirring occasionally, until softened, 7 to 8 minutes. Transfer mixture to refrigerator; chill 5 minutes.
Transfer onion mixture to a food processor; add shrimp, sausage, rice, scallions, creole seasoning, and filé powder. Pulse until finely chopped, 25 to 30 times. Shape mixture evenly into 4 (3/4-inch-thick, or 7 1/4 oz.) patties.
Heat oil in a large nonstick skillet over medium. Add patties; cook until cooked through, 6 to 8 minutes per side
Assemble the burgers: Spread 1 tablespoon Ketchup on each top and bottom bun half. Top each bottom half with 1 Patty and 4 okra pieces. Top with top bun halves.