Divide dough into two equal portions. On a lightly floured surface roll out each dough portion into a 10x13-inch oval. Stack dough ovals between sheets of waxed paper or parchment paper sprinkled with cornmeal; place on a large baking sheet. Freeze dough about 10 minutes or until firm but not frozen.
In a small saucepan heat 1/4 cup of the oil over medium heat. Add garlic; cook and stir for 30 seconds. Remove from heat; let garlic oil cool.
Brush both sides of eggplant slices with remaining olive oil. For a charcoal grill, grill eggplant on the lightly greased rack of an uncovered grill directly over medium-hot coals for 6 to 8 minutes or until eggplant is golden brown and tender, turning once halfway through grilling. (For a gas grill, preheat grill. Reduce heat to medium-high. Place eggplant on a lightly greased grill rack over heat. Cover and grill as directed.)
Remove pizza dough from freezer; discard waxed paper. Brush tops with some of the garlic oil; season to taste with salt and pepper.
For a charcoal grill, carefully slide dough, oiled sides down, onto the lightly greased rack of an uncovered grill directly over medium-hot coals. Grill about 4 minutes or until dough is light brown. Working quickly, carefully brush tops with garlic oil. Using tongs, carefully turn the crusts over. Immediately brush tops of crusts with more garlic oil, then top with mozzarella, eggplant, and tomatoes. Cover and grill about 3 minutes more or until cheese melts. Transfer pizzas to a cutting board. Sprinkle with basil and drizzle with balsamic vinegar. (For a gas grill, preheat grill. Reduce heat to medium-high. Place dough on lightly greased grill rack over heat. Grill as directed.)