Step 1

Peel the whole eggplants; coarsely chop the flesh and transfer to a large colander. Using a spoon, scoop the flesh from the eggplant halves, leaving a 1/4-inch-thick shell. Chop the scooped out eggplant flesh and add it to the colander. Toss the chopped eggplant with 2 tablespoons of kosher salt and let drain for 30 minutes, then rinse well. Working in handfuls, squeeze out as much of the water as possible. You should have about 4 cups of chopped eggplant.

Step 2

Meanwhile, bring a large pot of salted water to a boil. Set a wire rack on a baking sheet. Add the 6 eggplant shells to the pot and cook, gently poking them under to keep them submerged, until just tender, about 3 minutes. Using a slotted spoon, transfer the eggplant shells to the wire rack to drain and cool. Lightly oil a 9-by-13-inch baking dish and arrange the eggplant shells in it, cut sides up.

Step 3

In a large, deep skillet, heat 3 tablespoons of olive oil until shimmering. Add the onion and bay leaves and cook over moderate heat until the onion is softened, about 5 minutes. Add the chopped eggplant and wine and cook over moderate heat, stirring occasionally, until tender and just beginning to brown, 15 minutes. Add the veal and cook over moderately high heat, stirring and breaking up the meat, until cooked through and lightly browned, about 5 minutes longer. Transfer the eggplant filling to a bowl and stir in 1/2 cup of the pecorino, the eggs, bread crumbs, nutmeg, basil and mint. Season the filling with salt and pepper. Spoon the filling into the eggplant shells.

Step 4

Preheat the oven to 350°. In a medium saucepan, heat 1 tablespoon of olive oil. Add the garlic and cook over moderate heat until golden, about 1 minute. Add the chopped tomatoes and cook over moderate heat, stirring, until thickened, about 10 minutes. Season with salt and pepper. Spoon half of the sauce over the eggplants and sprinkle with the remaining 1/2 cup of grated pecorino. Bake the eggplant until browned and bubbling, about 35 minutes. Let cool slightly, then serve, passing the remaining tomato sauce on the side.