In a skillet, melt 1 tablespoon of butter over medium heat. Add the chopped onion, celery, red bell pepper, and minced garlic. Sauté until the vegetables are soft and fragrant, about 5 minutes. Remove from heat and let cool.
In a large mixing bowl, combine the sautéed vegetables with the mayonnaise, Creole mustard, Worcestershire sauce, Creole seasoning, and the lightly beaten egg. Mix well until all ingredients are combined.
Gently fold in the crushed Saltine crackers and the coarsely chopped crawfish tails into the mixture. Be careful not to overmix; you want to maintain some texture.
Using your hands, form the mixture into patties, about 1/2 inch thick. You should be able to make approximately 6 patties. Place them on a plate and refrigerate for about 30 minutes to help them firm up.
In a large skillet, heat the remaining 2 tablespoons of butter and 2 tablespoons of vegetable oil over medium heat. Once hot, add the patties to the skillet, cooking them for about 4-5 minutes on each side, or until golden brown and cooked through.
While the patties are cooking, whisk together the mayonnaise, chopped dill, dill pickle relish, lemon juice, Creole mustard, and Creole seasoning in a small bowl. This will be your delicious sauce for the burgers.
To assemble, place a cooked crawfish patty on the bottom half of each bun. Top with green lettuce, a slice of tomato, and a generous dollop of the creamy dill sauce. Place the top half of the bun on and serve immediately.