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.
