For the mayonnaise mixture: Combine the mayonnaise and garlic in a small bowl and season with salt and pepper. Refrigerate for at least 30 minutes before using.
For the roasted pepper relish: Combine the basil, vinegar, oil and peppers in a small bowl and season with salt and pepper. Let sit at room temperature while you assemble the sandwiches.
Preheat a grill on medium-high heat.
For the grilled eggplant: Brush the eggplant slices with oil and sprinkle with salt and pepper. Cook until grill marks appear, 1 to 2 minutes per side.
For the sandwiches: Brush the cut sides of the bread with oil and season with salt and pepper. Place on the grill cut-side down to warm through and crisp slightly, about 30 seconds.
Spread the bread with the mayonnaise mixture and top the bottom halves with the grilled eggplant, roasted pepper relish, mozzarella slices, arugula and basil. Cover with the tops of the bread. Place the sandwiches on the grill, weigh them down with a brick wrapped in foil and cook until pressed and the cheese has melted, flipping once, about 6 minutes. (They can also be cooked on a panini press.)