Bring a large pot of water to a rapid boil. Add the lobster and cook for 8 minutes. Remove from the water and set aside to cool slightly.
Preheat the oven to 450 degrees F (220 degrees C). When the lobster is cool enough, split it in half lengthwise using a large chef's knife. Remove the dark strip if present. Remove the tomalley to a bowl and mix with the bread crumbs, butter, salt, pepper and tarragon. Stuff this mixture on top of the lobster meat still in the shell and place lobster halves in a baking dish. Drizzle oil over the top.
Bake in the preheated oven until the stuffing is browned, about 20 minutes.