In food processor fitted with a metal blade, combine flour, yeast, rosemary, sugar, and pinch of salt. Pulse to mix. With machine running, pour in hot water and oil through feed tube. Process for 1 minute, or just until a ball forms. Continue to process for 35 to 45 seconds to knead dough.

With floured hands, shape into a smooth ball. Place in lightly oiled bowl. Cover and let rise in a warm draft-free place until doubled, about 1 hour. Punch dough down and proceed with recipe. Alternatively, place in a tightly sealed container and refrigerate for up to 8 hours or, if not using right away, tightly wrap and freeze for up to a month.

Grease barbecue and preheat to medium-high. Before rolling out, bring dough to room temperature. Using a lightly floured rolling pin and on a lightly floured surface, roll out dough into a 12 inch oval. Do not overwork or knead it. Have all your toppings ready before you place flatbread on grill.

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.

Transfer flatbread to cutting board. Scatter arugula overtop and drizzle with vinegar and oil. Season with salt and pepper to taste if you wish. Cut into wedges and serve.