Bring a large pot of salted water to a boil. Add the live lobsters and cook for about 8-10 minutes, until they turn bright red and are cooked through.
Using a slotted spoon, remove the lobsters from the pot and let them cool slightly. Once cool enough to handle, remove the meat from the claws, tails, and knuckles. Chop the meat into bite-sized pieces.
In a medium mixing bowl, combine the mayonnaise, lemon juice, Dijon mustard, chopped chives, salt, and black pepper. Mix well.
Add the lobster meat to the dressing and gently fold to combine, ensuring the lobster is well coated but not mushy.
Preheat a grill or frying pan over medium heat. Spread 1 tablespoon of butter on each side of the hot dog buns. Place the buns on the grill or in the pan and toast until golden brown.
Once the buns are toasted, fill each bun generously with the lobster salad mixture.
Serve immediately, garnished with additional chives if desired.