For the crust: Sprinkle the yeast over 3/4 cup warm (not lukewarm) water in a bowl.
In a stand mixer with the paddle attachment, add the flour and salt and, with the mixer running on low speed, drizzle in the olive oil. Keep going until it's mixed through. Next, pour in the yeast/water mixture and mix until just combined.
Coat a separate mixing bowl with a light drizzle of olive oil, tip the dough in and form it into a ball. Toss to coat the dough ball in the olive oil, then cover the bowl tightly with plastic wrap and allow the dough to rise for at least an hour, or up to 3 or 4 days.
Preheat the oven to 500 degrees F. Arrange the oven rack in the lowest position.
For the topping: Roll out the pizza dough on a lightly floured surface as thinly as possible Dough should be roughly 17 by 10 inches. Place on a large baking sheet. Drizzle lightly with the olive oil and sprinkle lightly with salt. Spread the fig spread all over the surface of the dough. Lay the slices of mozzarella all over the surface of the pizza crust. Sprinkle lightly with salt and pepper. Bake the pizza until the crust is golden and the cheese is bubbly, 12 to 15 minutes.
Remove from the oven and immediately drape the prosciutto slices over the hot pizza. Sprinkle generously with the arugula and Parmesan shavings. Cut into wedges or squares and serve immediately!