Melt 1 tablespoon butter in a medium skillet over medium-high heat. Add onion, celery, and red bell pepper. Cook, stirring occasionally for 4 minutes.
Add garlic and cook 1 more minute. Let mixture cool and transfer to a large bowl.
Add mayonnaise, mustard, Worcestershire sauce, Creole seasoning, egg, Saltine crackers, and crawfish tails. Stir to combine. Cover with plastic wrap and refrigerate 4 to 24 hours.
Make dill sauce. Combine all ingredients in a small bowl. Cover and refrigerate until needed.
Shape mixture into 6 patties or 10 smaller ones if making sliders.
Heat 1 tablespoon butter and 1 tablespoon oil in a nonstick skillet over medium heat. Add half the patties and cook for 4 to 5 minutes per side or 3 to 4 minutes for slider-sized patties.
Repeat with another tablespoon of butter and oil and remaining patties.
Layer lettuce, tomato slices, and a crawfish patty on each bun. Spread top of bun with dill sauce.
Notes
The crawfish patties can be shaped and formed and frozen for later use. Wrap well in plastic wrap and defrost in the refrigerator overnight before cooking them.