Bring 5 cups water to boil in large saucepan over high heat. Stir in shrimp and 1 tablespoon salt. Cover and let stand off heat until shrimp are firm and opaque, about 6 minutes, shaking saucepan halfway through. Fill large bowl halfway with ice and water. Transfer shrimp to ice bath and let cool for 5 minutes.
Drain shrimp and pat dry with paper towels. Cut half of shrimp into ½-inch pieces, coarsely chop remaining shrimp into pieces smaller than ½ inch.
Spread butter evenly on outer sides of buns. Toast outside of buns in 12-inch skillet over medium heat until deep golden brown, about 5 minutes. Transfer buns to serving platter.
Whisk mayonnaise, celery, chives, lemon zest, salt, and cayenne together in medium bowl. Add shrimp and gently toss to combine. Season with salt to taste. Line each bun with lettuce leaf, if using. Divide shrimp mixture among buns and serve with lemon wedges.