Season both sides of the eggplant slices with salt and pepper.
In a small bowl, whisk together the balsamic vinegar, olive oil, 1 teaspoon (5 ml / 0.2 fl oz) of fresh thyme, and minced garlic.
Brush this marinade generously over both sides of each eggplant slice.
Preheat your grill to medium heat (about 350-450°F / 175-230°C (450°F)).
Place the eggplant slices directly on the grill rack.
Grill for 10 to 12 minutes, flipping once, until the eggplant is tender and has beautiful grill marks.
During the final 3 minutes of grilling, place the tomato slices on the grill to warm through.
Top each eggplant slice with a sprinkle of Parmesan and mozzarella cheese and close the grill lid to allow the cheese to melt.
To serve, carefully remove the eggplant and tomatoes from the grill.
Place an eggplant slice on a plate, top with a grilled tomato slice, and serve with a side of warmed pasta sauce for dipping or drizzling.
Garnish with additional fresh thyme, if desired.