Pour 12 cups water into a large pot. Cut 2 lemons in half and squeeze juice into pot through a sieve or your hand to catch the seeds; toss in spent lemon halves. Season water with kosher salt; bring to a boil.
Remove pot from heat and add 1 lb. large shrimp, peeled, deveined. Let sit, uncovered, until shrimp are pink and cooked through, 1–2 minutes. Using a slotted spoon, transfer shrimp to a medium bowl of ice water. Drain shrimp. Coarsely chop, then place in a large bowl.
Finely grate zest from remaining 1 lemon to yield 2 tsp., then cut lemon in half and squeeze out 1 Tbsp. juice. Reserve any remaining juice for another use.
Heat 4 Tbsp. salted butter, cut into small pieces, and 1 garlic clove, finely grated, in a small microwave-proof bowl in a microwave until butter is melted, about 30 seconds. Pour half over shrimp and toss to coat. (Don’t worry if butter solidifies.) Add lemon zest and juice, 1 celery stalk, finely chopped, ⅓ cup ¾”-long pieces chives, ⅓ cup mayonnaise, and 2 tsp. Dijon mustard and toss to coat. Taste and season with salt if needed. Chill shrimp mixture while you toast the buns.
Heat a dry large skillet, preferably cast iron, over medium. Brush insides of 6 hot dog buns, split, with remaining garlic butter. Working in batches, toast buns, cut side down, in pan until golden brown underneath, about 30 seconds per batch.
Line insides of buns with 1 small head of butter lettuce, leaves separated, and fill with chilled shrimp mixture.