Heat grill or grill pan to high. Coat the surface of the grill with oil or spray to avoid sticking. Place fruits cut side down on grill and cook 3 to 4 minutes until warmed through and evenly marked.
Heat the remaining ingredients (for the glaze) in small pot over medium-high heat or place on grill next to fruit. When the sauce reduces, it will become syrup-like in no time - less than a minute. If you burn food often or get distracted easily, cook it over lower heat. Otherwise, let it rip. When the balsamic sauce coats the back of a spoon and reduces by half, remove the pan from the heat and discard the ginger chunk.
Arrange fruits on a platter and drizzle liberally with the glaze.