Cut the eggplants in half lengthwise, including the stems. Using a sharp paring knife, scoop out some of the flesh, leaving a shell that is 1/3 inch thick and reserving the scooped-out eggplant. Place the eggplant shells snugly in a single layer in a large (10-by-14-inch) rectangular baking dish. Brush the eggplant all over with 1/4 cup olive oil (use it all!) and sprinkle with 1 1/2 teaspoons salt and 3/4 teaspoon pepper. Pour 1/4 cup water in the baking dish, cover tightly with aluminum foil, and bake for 30 minutes, until the shells are tender when pierced with a small knife. Remove from the oven, discard the foil, and leave the oven on.
Meanwhile, for the lamb filling, in a large (12-inch) skillet, heat 3 tablespoons olive oil over medium heat. Add the onion and saute for 5 minutes, tossing occasionally, until tender. Cut the reserved scooped-out eggplant into 1/2-inch dice (you should have about 2 cups). Add the eggplant to the onion and cook for 5 minutes, adding more oil if it seems dry. Add the garlic and cook for 30 seconds, until fragrant. Add the ground lamb and cook, mashing lightly with a fork, for 5 to 7 minutes, until no longer pink. Add the oregano and cinnamon and cook for one minute. Add the wine, tomatoes, marinara, parsley, 2 teaspoons salt, and 1 teaspoon pepper. Bring to a boil, lower the heat, and simmer uncovered for 25 to 30 minutes, until the liquid evaporates. Spoon the filling into the eggplant halves in the baking dish. (You may have a little extra filling.)
In a medium bowl, mash the ricotta, goat cheese, egg yolks, mint, 1 teaspoon salt, and 1/2 teaspoon pepper with a fork. With a spoon, place large dollops of the ricotta mixture on the lamb filling and bake for 30 to 40 minutes, until the filling is hot and the topping is lightly browned. Serve warm, sprinkled with extra mint.