If using whole lobsters, bring a large pot of salted water to a boil. Cook the lobsters for about 5 minutes, then transfer them to a bowl of ice water to cool.
Once cooled, remove the meat from the shells and chop it into bite-sized pieces.
In a mixing bowl, combine the chopped lobster meat, mayonnaise, lemon juice, Dijon mustard, Old Bay seasoning, salt, and pepper. Gently mix until the lobster is well coated. Adjust seasoning to taste.
Heat a skillet or griddle over medium heat. Add butter and let it melt, coating the cooking surface.
Place the hot dog buns in the skillet and toast them until golden brown on both sides, about 2-3 minutes.
Remove the toasted buns from the skillet and fill each bun with the lobster mixture, generously packing it in.
Garnish with fresh chives or parsley, if desired, and serve immediately.