Bring 5 cups water to boil in large saucepan over high heat. Stir in 1½ pounds shrimp, peeled, deveined, and tails removed and 1 tablespoon table 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 2 tablespoons salted butter, softened, evenly on outer sides of 4 New England–style hot dog buns. Toast outside of buns in 12-inch skillet over medium heat until deep golden brown, about 5 minutes. Transfer buns to serving platter.

Melt remaining 4 tablespoons butter in now-empty skillet over medium heat. Continue to cook, stirring occasionally, until butter stops spattering. Reduce heat to medium-low and stir in shrimp, 1 minced small shallot, and ¼ teaspoon salt. Cook, stirring occasionally, until shrimp is heated through, 2 to 3 minutes.

Divide shrimp mixture among buns. Drizzle with any remaining butter from skillet and sprinkle rolls with 2 teaspoons minced chives. Serve with lemon wedges.