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1
Steam or boil the lobsters until the shells are uniformly bright red and the tail meat is just opaque when pulled free — this is the ceiling of acceptable cooking. Extract all claw, knuckle, and tail meat. Cut into generous, irregular chunks; do not mince. Reserve, covered, under refrigeration if making the Maine version, or hold loosely covered at room temperature if making the Connecticut version.
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2
For the Maine version: combine the cold lobster meat with the mayonnaise, celery, and lemon juice. Turn gently until the meat is barely coated — this is where the dish lives or dies. Excess mayonnaise masks the lobster; insufficient dressing leaves it dry. The mixture should look glossy, not pale or heavy. Season with fine sea salt. Hold cold until the buns are ready.
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3
For the Connecticut version: melt the butter in a small pan over low heat and keep it warm. Immediately before filling, tumble the lobster chunks through the warm butter until each piece is glistening and heated through. Season with fine sea salt. The meat should steam lightly when turned; it must not sizzle or colour.
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4
Butter the flat outer faces of the top-split buns with softened butter. Place face-down on a flat griddle or heavy pan over moderate heat. Press lightly. Cook until each face is uniformly golden and the surface feels lightly crisp under a fingertip — the interior should remain soft.
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5
Open the griddled buns and fill generously with the dressed lobster — Maine cold, Connecticut warm. Serve immediately; both versions deteriorate within minutes of assembly.