Sprinkle eggplant slices with kosher salt and let sit for 10 minutes or so. This helps draw out any bitterness that may be in the eggplant. You should see little drops of water on the surface of the eggplant after it sits - brush off salt and liquid with a paper towel.
Brush eggplant slices on both sides with olive oil. Sprinkle with black pepper and garlic powder. Grill on high heat for 10-15 minutes, flipping once, until tender. (flipping only once and leaving the slices alone in between will help you get those nice grill marks)
Preheat broiler if you used an indoor grill.
Remove eggplant from heat. Pair up eggplant slices of similar size. Top one slice from each pair with half the red peppers, then slices of fresh mozzarella. Put on a baking pan under the broiler or back on the grill with the cover down for 3 minutes or until cheese is melty. Remove from heat.
Layer with 1 tbsp basil, the second slice of eggplant, the remaining red peppers, and the other 1 tbsp basil.
Drizzle with the best balsamic vinegar you have, or balsamic reduction, and enjoy. Garnish with basil leaves if desired.