How to Make It
Cut eggplants crosswise into 20 slices about 3/4 inch thick; place in a 13- x 9-inch baking dish.
Whisk together 1/4 cup oil, 1/4 teaspoon sea salt, lemon juice, and shallot. Pour mixture over eggplants, turning to coat. Marinate 30 to 45 minutes. Meanwhile, soak bamboo skewers in water 30 minutes.
Preheat grill to medium-high heat (350° to 400°). Thread tomatoes on skewers; brush with 1 tablespoon oil. Grill 8 minutes, turning once.
Combine grilled tomatoes, garlic, vinegar, red pepper, 1/2 teaspoon sea salt, and remaining 2 tablespoons oil in a food processor. Pulse until a chunky sauce forms.
Grill eggplant slices 5 minutes. Turn over, and sprinkle with remaining 1/4 teaspoon sea salt. Top each eggplant slice with 1 mozzarella cheese slice, and grill 5 minutes or until eggplants are tender and cheese begins to melt.
Pour tomato sauce onto a serving platter. Top with eggplants, and sprinkle with Pecorino-Romano cheese and basil.