Preheat the grill to high or the oven to 450˚.
Rinse the figs and pat dry. With a paring knife cut a small slit in the bottom of each fig. Insert a cube of cheese and with your fingers gently squeeze the opening closed. Wrap one slice of prosciutto around each fig, rolling the fig to blanket it completely. Repeat with the remaining figs and place on the grill. With the lid close, grill until the figs are browned on one side, about 4 minutes. Flip the figs and repeat.
Alternatively, preheat the broiler to high and place the figs in a roasting pan. Broil until browned, about 2 minutes. Flip the figs and broil another 2 minutes or until golden brown and soft.
Place the warm figs on a serving platter and drizzle with the oil and vinegar. Sprinkle with the salt to taste. Serve immediately.