If your shrimp is not pre-cooked: bring a large pot of water to a boil. Throw in the raw shrimp and cook 2-3 minutes until pink and opaque. Drain and rinse under cold water to stop cooking, then toss the shrimp into a bowl of ice water for 10 minutes. Dry the shrimp well and chop.
Add the shrimp to a large bowl. Stir in the mayonnaise, pickles, celery, pickle brine, hot sauce, Old bay, dill, garlic, onion, cayenne pepper, and a pinch of salt and pepper until the shrimp is very creamy and fully coated.
Melt the butter in a large skillet over medium heat until browned, about 2-3 minutes. Set each bun in the skillet and toast for 1 minute per side, until golden.
Stuff a lettuce leaf into each roll, then add gobs of the chopped shrimp mixture. Garnish with fresh chill and serve with potato chips!