How to Make It
Grease the bottom of a large bowl with olive oil. Pour the warm water into the bowl of a stand mixer fitted with the dough hook. Evenly sprinkle the yeast and sugar over the water. Mix at low speed until combined, 30 seconds. Add the flour, rosemary, 1/2 cup of the olive oil and 4 teaspoons of salt and mix at medium speed until a dough forms, about 5 minutes; it will be a little sticky. Form the dough into a ball and transfer to the greased bowl. Cover tightly with plastic wrap and let stand at room temperature until it doubles in bulk, about 2 hours.
Punch the dough down in the bowl and form it into a loose ball. Cover with plastic wrap and let stand at room temperature for 1 hour.
Meanwhile, in a small, heat-proof bowl, cover the figs with boiling water. Let stand until softened, 1 hour. Drain the figs and slice them 1/4 inch thick.
Lightly grease a 9-by-13-inch rimmed baking sheet. Transfer the dough to the prepared pan and, using your fingertips, press it out to fill the pan. Cover with plastic wrap and let rise at room temperature for 30 minutes.
Preheat the oven to 375°. Arrange the fig slices all over the dough; gently press them in. Drizzle with 2 tablespoons of the olive oil and season with salt and pepper. Sprinkle the cheese over the top. Bake for 25 to 30 minutes, rotating the pan halfway through, until golden and firm. Let cool slightly on a rack. Drizzle with the remaining 2 tablespoons of olive oil and serve warm or at room temperature.