Arrange the puff pastry dough on a baking sheet and bake until golden brown, about 15 minutes.
Remove the pastry from the oven and spread it with pesto. Next cover the pastry with overlapping slices of tomatoes. Season with salt and pepper, to taste. Cover the tomatoes with the grated Parmesan and return it to the oven. Bake until the cheese has melted, about 5 to 10 minutes. Remove the tart from the oven and cut it into 6 equal pieces. Arrange the tarts on serving plates and drizzle with balsamic reduction.
To make Balsamic Reduction: Combine the balsamic vinegar and sugar in a small sauce pot and bring to a boil over medium heat. Cook until the liquid is reduced by half and has a syrupy consistency, about 30 to 45 minutes. Remove from heat and allow reduction to cool before using.