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.