Combine the mayonnaise and garlic in a small bowl and season with salt and pepper, to taste. Refrigerate for at least 30 minutes before serving.
Heat a grill to high.
Brush the eggplant on both sides with oil and season with salt and pepper, to taste. Grill until golden brown on both sides and just cooked through, about 4 minutes per side. Brush the peppers with oil and season with salt and pepper, to taste. Grill until charred on all sides, remove, place in a bowl, cover and let steam for 10 minutes. Remove the skin and the seeds.
Put bread on the grill, cut side down, and grill for 30 seconds to warm through and crisp slightly. Spread the bread with the mayonnaise mixture and top the bottom half with eggplant, peppers, cheese, basil and arugula. Cover with the top of the bread and serve.