To make Tapenade:
In a food processor or blender, combine the olives, sun-dried tomatoes, capers, garlic, vinegar and 2 teaspoons of olive oil. Process to a chunky puree, scraping down the sides as needed. Set aside. (The tapenade can be made ahead and refrigerated until needed.)
To make sandwich:
Preheat a gas grill to medium or prepare a charcoal grill.
Spread the tapenade evenly over one side of each slice of bread. Set aside.
Brush the eggplant slices with remaining olive oil on each side and place on the grill. Cook until golden brown and tender, about 5 to 6 minutes per side.
Divide the eggplant slices between four slices of bread, arranging the eggplant over the tapenade. Top the eggplant with tomato slices, mozzarella cheese and then sprinkle with chopped basil.
Firmly press a second slice of bread on top of each to make sandwiches.
Brush each side of the bread of the sandwiches with melted butter. Grill the sandwiches until the filling is hot and the bread is browned and crisp, turning them carefully with a spatula, 2 or 4 minutes per side.
Cut each sandwich in half to serve.