Arrange 2 flatbreads side by side and add pesto (divide evenly between 2 breads). Lay on slices of tomatoes, then top with sliced mozzarella.
Heat grill over medium heat. Lay flatbreads directly onto the grill and close grill. Check after 2 minutes, and watch carefully as they might burn quickly if your heat is too high. Remove from grill when cheese is melted and they are toasted on the bottom (does NOT take a long time, just a few minutes), set aside.
Prepare your glaze: add balsamic vinegar and sugar to a small sauce pot. Whisk to combine and bring to a boil. Simmer, stirring occasionally for about 10 minutes. Let cool for 3-5 minutes and sauce will thicken.
Drizzle balsamic glaze over the flatreads, cut into fourths, and serve immediately.