Preheat grill. You will need medium to high heat for grilling. The key is to crisp the prosciutto quickly and leave the blue cheese just melted with the inner part of the fig cool in temperature.
Cut the figs in half and place a piece of blue cheese on each fig half. Wrap the prosciutto around each fig half, covering the cheese. The ends of the prosciutto should overlap.
Grill each piece until the prosciutto begins to color and crisp, about 2 minutes on each side. Remove from grill, lightly drizzle with extra virgin olive oil and season with sea salt and pepper. Serve warm.