Brush top of flatbread with oil. Place, oiled side down, on grill and barbecue with lid open until bottom side is golden brown, about 3 minutes. Carefully rotate flatbread with tongs to prevent burning. Brush top of flatbread with oil and turn dough over. Immediately arrange beet slices, figs, and cheese on top. Continue to grill with barbecue lid open for 3 more minutes, until cheese is melted and bottom of flatbread is golden. Rotate flatbread frequently, checking to see that the undersides are not charring. If flatbread is turning dark too quickly, turn off grill on one half of barbecue; slide flatbread to side of grill without the flame and close lid. Grill just until toppings are hot.