To prepare the eggplant, preheat your grill to medium-high heat. Brush both sides of the eggplant slices with olive oil and sprinkle with salt. Grill the eggplant slices for 4-5 minutes on each side until tender and slightly charred.
While the eggplant is grilling, brush the cut sides of the ciabatta with olive oil. Place the ciabatta on the grill, cut side down, and grill until golden and toasted, which should take about 2-3 minutes.
Next, spread marinara sauce over the grilled sides of the ciabatta. Layer the grilled eggplant slices on top of the marinara and add slices of fresh mozzarella over the eggplant. Close the grill lid and cook until the mozzarella is melted and bubbly, about 2-3 minutes.
Remove the toasts from the grill and sprinkle with grated Parmesan cheese. Top with plenty of fresh basil leaves. Slice and serve immediately.