To par-cook the lobsters: Bring a large pot of salted water to a boil. Add the lobsters and cook about 7 minutes; they will be about three-quarters done. Drain well and let cool. Cut in half lengthwise.
Heat a charcoal or gas grill to high for direct grilling. Brush the cut-side of the lobsters with some oil and sprinkle with salt and pepper. Grill, cut-side down, until slightly charred. Flip and continue grilling until lightly charred and heated through. Remove the lobsters from the grill and let cool slightly.
Brush the insides of the hot dog buns with some oil and sprinkle with salt and pepper. Grill until just toasty.
Remove the tail and claw meat from the lobsters. You can cut them, but just enough to give you about 2 very large pieces per bun. Drop them into the Lemon-Tarragon Clarified Butter and give them a good soaking. Spoon the lobster into the toasted buns and serve immediately.
Lemon-Tarragon Clarified Butter:
Melt the butter slowly in a medium saucepan over low heat. Let it sit for a bit to separate. Skim off the foam that rises to the top, and gently pour into a bowl, leaving behind the milk solids, which have settled to the bottom of the pan.
Put the clarified butter in a large saute pan over low heat. Add the lemon zest, lemon juice and tarragon, and season with salt, pepper and cayenne if desired.