In a large bowl toss the eggplant slices with 1/4 cup kosher salt. Using your fingertips, evenly distribute the salt on both sides of each slice. Place the eggplant in a colander set over a large bowl or sheet pan to catch juices. Set aside for 1 to 2 hours.
Preheat oven to 450 degrees F. Line two 18 by 13-inch baking sheets with heavy-duty foil.
Rinse the eggplant under cold water and dry thoroughly with paper towels.
Arrange the eggplant slices in a single layer on one baking sheet. Brush olive oil on 1 side. Brush with balsamic vinegar and sprinkle with freshly ground pepper.
Drizzle the other pan with 2 tablespoons olive oil. Arrange the tomato slices in a single layer, season with salt and pepper, then sprinkle with the minced garlic.
Roast the tomatoe slices until they are very soft and just beginning to brown, 10 to 15 minutes, without turning. Bake the eggplant slices until they are tender and well browned, 20 to 30 minutes, without turning. Let the tomato and eggplant slices cool until you can handle them with your fingers.
Make a short stack starting with eggplant on the bottom, 1 slice mozzarella, and then tomato. Top with another piece of eggplant. Warm stacks in the oven until mozzarella lightly melts. Drizzle with olive oil and serve.