In a large bowl, sprinkle yeast over 3/4 cup warm (110°) water. Let stand until softened, about 5 minutes. Stir in salt and 1/3 cup olive oil.
If using a dough hook, add 2 1/2 cups flour and mix until moistened. Beat on high speed until dough no longer feels sticky and pulls cleanly from bowl, about 10 minutes. If dough is still sticky, beat in more flour, 1 tablespoon at a time.
If mixing by hand, add 2 1/2 cups flour and beat until dough is moistened and slightly stretchy. Scrape dough onto a lightly floured board. Knead until dough is smooth, elastic, and no longer sticky, 10 to 12 minutes; add flour as required to prevent sticking.
Cut dough into 4 equal pieces. Place on floured board and cover with plastic wrap to prevent drying. Let rest 10 minutes. Knead 1 piece of dough to expel air. On a floured board, roll out to make an 11-inch round. Put round on a 12- by 15-inch baking sheet and prick dough all over with a fork.
Bake on bottom rack of a 500° oven until pale golden, about 3 minutes. Slide crust onto a rack to cool. Repeat to shape and bake remaining dough.
To reheat, place baked crusts in a single layer directly on racks in the oven until hot, about 1 minute.
Transfer crusts to plates and quickly arrange prosciutto, figs, and arugula equally on each. Sprinkle evenly with remaining oil, rosemary, and pepper.