Whisk together first 5 ingredients until blended.
Drizzle 3 tablespoons vinaigrette over fig halves, and reserve remaining dressing
Grill figs, covered with grill lid, over medium-high heat (350° to 400°) for 2 to 3 minutes on each side or until tender. Remove from heat and cut fig halves in half. Cover and chill 30 minutes.
Combine chèvre and next 3 ingredients in a food processor. Process until well blended, stopping to scrape down sides.
Toss greens with remaining vinaigrette. Spoon chèvre mixture in the center of each plate. Top with figs, prosciutto, and greens.