Preheat a grill or broiler.
Cut the eggplants lengthwise into slices about 3/8 inch thick. Cut these slices into long, 1-inch-wide strips. Brush the strips using 3 to 4 tablespoons of the olive oil and arrange them on a grill rack or broiler tray. Grill or broil them, turning once, until golden, 3 to 5 minutes per side. Remove and set aside until cool enough to handle, about 10 minutes.
Arrange a layer of eggplant in the bottom of a dry-sterilized, wide-mouthed 1-quart jar with a lid. Sprinkle with some of the salt, pepper, garlic and thyme leaves. Continue layering in the same manner, packing tightly, until all of the eggplant, garlic and thyme have been used. Combine the remaining olive oil and the vinegar and pour it over the layers. Cover with the lid.
Place the jar in the refrigerator. Turn the jar upside down every other day for a week to allow the liquid to penetrate all the layers. At the end of the week, the eggplant will be ready to eat, but the flavor will continue to develop with age. The preserved eggplant will keep for up to 6 months in the refrigerator.