Sprinkle the eggplant slices with salt on both sides, then place on a rack and set aside for 30 minutes. The salt will draw out excess moisture and bitterness.

While the eggplant is sitting, make the basil oil. Combine the garlic, basil, lemon juice and olive oil in a food processor or blender and process on high until smooth, about 1 minute. Season with salt and pepper, then set aside.

Use paper towels to blot the excess moisture and salt from the eggplant, then drizzle with olive oil and rub to coat on all sides. Sprinkle with freshly ground black pepper to taste.

Preheat a grill to medium-high heat. Use paper towels to pat the barramundi fillets very dry. Drizzle with olive oil and rub to coat on all sides, then season both sides with salt and pepper. Grill the barramundi and eggplant pieces until nicely charred on the outside and the barramundi is opaque throughout.

Place a piece of eggplant in the center of a plate, then spoon a little bit of the basil oil over top. Stack a piece of tomato on top, season with salt and pepper, then drizzle with more basil oil. Top with a piece of fish, drizzle with more basil oil, then repeat with one more piece of each. Use up the remaining fish, tomato and eggplant to create three more stacks, then spoon a generous amount of basil oil over each. Finish with a light drizzle of aged balsamic or balsamic reduction around the perimeter of each plate. Serve immediately.