Cut each fig in half vertically, cutting to, but not through, base of fig. Spread slightly apart. Place 1 cube of cheese inside each fig; gently close to seal. Wrap one prosciutto strip around each fig.
Combine vinegar and next 3 ingredients (through salt) in a small saucepan; bring to a boil. Reduce heat, and simmer until thick, about 10 minutes. Cool.
Place figs on grill rack coated with cooking spray; grill 6 minutes, turning occasionally, until prosciutto is crisp. Drizzle each fig with balsamic syrup.