In a large bowl, combine mayonnaise, Dijon mustard, lemon juice, celery, and chives.
Gently fold in the chopped lobster meat until coated. Taste and adjust seasoning.
Heat a large skillet over medium heat. Brush the insides of the buns with melted butter, then place them cut side down in the skillet. Toast for about 2 minutes, until golden brown.
Spoon the lobster mixture into warm split-top brioche buns or regular hot dog buns. Sprinkle Old Bay over the top. Serve immediately.