Season the quartered pieces endive and readicchio with salt and pepper. Wrap each section with a half slice of ham. Brush the wrapped sections with extra-virgin olive oil and grill 7 to 8 minutes, turning every couple of minutes, until ham is crispy and lettuce sections are tender.
Place figs and vinegar in a pot and bring to a boil, then reduce heat to simmer. Reduce the vinegar down to a few tablespoons, until the fig pieces are soft and vinegar is thick, 5 to 7 minutes.
Arrange the lettuce on individual plates or a serving dish and spoon the fig and balsamic reduction over the lettuce, drizzling back and forth over the plate.