Pour Balsamic Syrup ingredients into small saucepan, and simmer over medium-high heat until reduced by half, 10 to 12 minutes. Cool to room temperature. Use 3 tablespoons syrup for recipe, saving remaining syrup in refrigerator for another use.
Preheat oven 425°F. Grease 2 cookie sheets.
Divide dough into 4 pieces. Roll out 2 dough pieces on each cookie sheet into 8-inch rounds. Brush 1/2 tablespoon olive oil on each dough round.
Divide chicken, peaches and red onion evenly among dough rounds.
Bake 10 to 12 minutes or until crust is golden brown. Remove from oven, and top with basil, blue cheese and arugula. Drizzle each pizza with part of the 3 tablespoons balsamic syrup, and season with salt and pepper.