Preheat a grill to medium. Brush the eggplant with the olive oil and season with salt and pepper. Grill, turning once, until the eggplant is tender, about 6 minutes. Coarsely chop and transfer to a bowl. Toss with the caponata and mozzarella.
Divide the pizza dough into 2 pieces. Roll out the dough into two 7-by-10-inch rectangles. Place the dough directly onto a well-oiled grate, cover and grill until the dough begins to bubble and grill marks appear, about 3 minutes. Flip and top with the eggplant mixture, then cover and grill until the cheese is melted and the crust is golden, 3 to 4 minutes. Cut each pizza in half.