Preheat a gas grill or prepare a fire in a charcoal grill.
2 medium eggplant
Trim the eggplant and cut lengthwise into 1/4-inch-thick slices, discarding the first and last slices from each one; you should have 12 slices.
About 3 tablespoons extra-virgin olive oil
Lay the slices on a baking sheet and lightly brush on both sides with olive oil, using about 3 tablespoons oil.
Place on the grill and cook, turning once, until golden brown and soft, about 2 minutes on each side; return the slices to the baking sheet as they are cooked.
In a small bowl, combine the bread crumbs, thyme, and tomato sauce.
Lay the eggplant slices out on a work surface, with a narrow end toward you.
Divide the bread crumb mixture among them, using a scant 1/4 cup for each and placing it on the lower half of each one.
1/2 cup freshly grated Parmigiano-Reggiano
8 ounces fresh mozzarella (cut into 12 thin slices)
Sprinkle with the grated Parmigiano and lay the slices of mozzarella on top.
Fold the tops of the eggplant slices over to create little packages and transfer to a clean baking sheet.
Refrigerate for at least 1 hour, or as long as overnight, to marry the flavors.
Preheat the gas grill again or prepare another fire in the charcoal grill.
About 2 tablespoons extra-virgin olive oil
Carefully brush the eggplant packets on both sides with the remaining 2 tablespoons olive oil.
Place them gently on the hottest part of the grill and cook, unmoved, for 2 minutes, or until nice grill marks appear on the first side.
Gently flip over with a large spatula and cook for 2 more minutes, or until marked on the second side and hot throughout.
12 fresh basil leaves, cut into chiffonade (thin slivers)
Carefully transfer the packets to a platter and sprinkle with the basil.
Serve hot or at room temperature.