How to Make It
Preheat the oven to 400° and lightly oil a baking sheet. Arrange the eggplants, cut side down, on the baking sheet and bake for 20 minutes, or until softened but not falling apart.
Meanwhile, bring a small saucepan of salted water to a boil. Add the potato and cook until crisp-tender, about 5 minutes; drain. Heat 1 tablespoon of the olive oil in a large skillet. Add the onion and cook over moderately high heat for about 5 minutes, stirring occasionally, until softened. Add the lamb and cook, stirring occasionally to break up the meat, until no liquid remains and the lamb is beginning to brown, about 10 minutes. Add the potato and stock and cook over moderate heat until the mixture is dry and the potato is tender, about 8 minutes.
Heat the remaining 1 tablespoon of olive oil in a large nonstick skillet. Add the eggplants, cut side down, and cook over high heat until browned, 1 to 2 minutes. Transfer to a work surface and let cool. Carefully scoop the flesh out of each eggplant, leaving a 1/4-inch-thick shell. Chop the eggplant flesh. Add the eggplant and tarragon to the lamb mixture and season generously with salt and pepper.
Lightly oil a large baking dish. Arrange the eggplant shells in the dish and fill with the lamb mixture, mounding slightly. Sprinkle with the Parmesan and bake for about 10 minutes, until the filling is heated through and the cheese is melted. Preheat the broiler and broil for about 1 minute, until lightly browned. Transfer the eggplants to a platter and serve.