Thaw the lobster tails overnight in the refrigerator or in a bowl of cold water for 30 to 60 minutes, changing the water once or twice.

Bring a large pot of water to a gentle boil. Add kosher salt and squeeze in half the lemon. Add lobster tails and cook 5 to 8 minutes until shells are bright red and meat is opaque. Avoid a hard boil to keep the meat tender.

Transfer tails to a plate and cool until easy to handle. Use kitchen shears to cut down the top of each shell and pull out the meat. Roughly chop into generous bite-size pieces.

In a mixing bowl, combine mayonnaise, melted butter, lemon juice, celery, chives, Old Bay if using, salt, and pepper. Stir to combine.

Fold lobster meat gently into the dressing until lightly coated. Taste and adjust lemon juice or salt as needed.

Spread softened butter on the outside of each bun. Toast in a skillet over medium heat until golden brown on both sides.

Fill warm toasted buns with chilled lobster salad. Finish with extra chives and a squeeze of lemon if desired. Serve immediately.