Heat a grill or grill pan to medium-high heat.
Thinly slice the eggplant in discs; salt and drain in a colander, 15-20 minutes.
Warm the oil with garlic and rosemary to infuse. Brush the eggplant on both sides and sprinkle with black and red pepper. Grill the eggplant for 2-3 minutes on each side until marked and tender. Cool to room temperature.
Season the sliced tomatoes with salt and pepper. On a platter, scatter a bed of arugula and over it layer the tomatoes with eggplant, basil and mozzarella. Drizzle the salad with the remaining garlic oil and top with speck.