This is a dish whose flavors improve with time, so make it when you have your grill going anyway. Grill the eggplant when the coals are dying, assemble the recipe, and it will be ready for a meal the next day.
Wipe eggplants and slice lengthwise or diagonally into pieces a little less than 1 cm thick. Discard the end pieces that are mostly skin. Brush each piece on both sides with oil and grill over the coals. (You can also fry the eggplant slices instead of grilling them, or even broil them, if you prefer.) To make crisscross grill marks, turn each slice just once at a 45-degree angle. Cook on both sides until eggplant is nicely colored and tender. As you work, make a layer of cooked eggplant in a non-corrosive dish. Season with salt and freshly ground pepper, scatter garlic and basil leaves on top and splash a little vinegar over all. Add the next layer of eggplant and continue until everything is used. The warmth from the eggplant will bring out the flavors of the garlic and the basil. It will smell wonderful.
Cover the dish and refrigerate for several hours or overnight. Bring to room temperature before serving.