To prepare the balsamic reduction, in a small saucepan over medium-high heat, bring balsamic vinegar to a simmer. Reduce heat to medium-low, and simmer until the vinegar has reduced to 1/4 cup, about 20 minutes. Set aside and let cool.
Brush both sides of bread slices with olive oil. Brush cut side of peaches with olive oil. Preheat grill to medium. Grill bread until golden and crisp, about 2 minutes on each side. Grill peaches until tender, about 3 minutes on each side.
Remove bread and peaches from grill. Cut peaches into small slices. Top bread with goat cheese, arugula, prosciutto, and sliced peaches. Drizzle with balsamic reduction and extra olive oil. Serve immediately.