Bring the water and the kosher salt (for boiling) to a rolling boil in a large pot over high heat, 8–10 minutes.
Add the live lobster headfirst, cover, and return to a boil. Boil 8–10 minutes for 1–1.25 lb lobsters, until shells are bright red and the tail meat is opaque and firm.
While the lobsters cook, fill a large bowl with the ice and enough cold water to make an ice bath.
Transfer the cooked lobsters to the ice bath to stop cooking; cool until just cold, 10–15 minutes, then drain well.
Twist off claws and tails; crack shells and remove claw, knuckle, and tail meat. Discard shells. Cut the lobster meat into 0.75-inch pieces.
In a mixing bowl, gently fold together the lobster meat, celery, mayonnaise, and lemon juice until just coated. Season with the kosher salt (for seasoning) and black pepper. Cover and chill 10–15 minutes to let flavors meld.
Spread the outsides of the New England-style split-top hot dog buns with the unsalted butter. Griddle in a skillet over medium heat until the sides are golden, 1–2 minutes per side.
Fill each bun evenly with the chilled lobster salad, mounding generously. Serve immediately.