Place balsamic in a small pot on medium-high. Let simmer until reduced and noticeably thick. Remove from heat and let cool. Sauce will continue to thicken as it sits.
Roll out pizza dough and place on a floured baking sheet.
Preheat an outdoor grill for high heat. Lightly oil grill grate.
Place pizza dough on grill. Dough will start to puff up. Once the bottom of the crust has lightly browned, turn dough over.
Working quickly, brush the dough with olive oil. Top with pizza sauce, fresh tomatoes, and mozzarella. Reduce heat to low. Close lid and cook until cheese has melted.
Remove pizza from grill and top with fresh basil. Drizzle with balsamic reduction and serve.