Prepare a grill or grill pan for cooking over medium heat. Stir together the olive oil and garlic in a small bowl.
Slice the eggplant into eight 1/2-inch-thick rounds. Slice each of the tomatoes into four 1/2-inch-thick rounds. Put the slices on a baking sheet, and brush them on both sides with the garlic oil; continue brushing until all the oil is used, being especially generous with the eggplant. Sprinkle both sides with salt. Let sit 5 minutes to allow the eggplant to fully absorb the oil and salt.
Mix the Parmesan and chopped basil together in a small bowl.
Lightly oil the grill grates. Grill the bread slices until they have developed grill marks and some char, about 1 minute per side. Remove from the grill, drizzle with olive oil and set aside.
Grill the eggplant and tomato rounds, covered, until grill marks appear, about 4 minutes. Flip the eggplant rounds with a spatula, and place 1 tomato round on top of each. Top each stack with some of the Parmesan-basil mixture and the mozzarella. Grill, covered, to finish cooking the eggplant and to melt the mozzarella, about 4 minutes.
Garnish the eggplant Parmesan stacks with torn basil leaves and drizzle with balsamic vinegar. Serve with the grilled bread on the side.