Make the dough: In a large bowl, combine warm water with yeast and sugar. Let sit for about 5 minutes until foamy. Add flour, salt, and olive oil. Stir together until shaggy, then knead on a floured surface for about 8 minutes until smooth and elastic (or use a stand mixer with dough hook for 5-6 minutes). Place in an oiled bowl, cover, and let rise for 1 hour or until doubled in size.
Prep toppings while dough rises: Tear mozzarella into small pieces. Slice figs into quarters or eighths depending on their size. Shred or tear the prosciutto into bite-sized strips. Set aside.
Heat oven and pan: Preheat your oven to 475°F with a pizza stone inside (or use a heavy baking sheet) for at least 20 minutes before you're ready to bake.
Shape the dough: Punch down risen dough. Divide into two equal portions if making two pizzas, or keep as one large round. Roll out on a lightly floured surface to about ¼-inch thick — not too thin since we want some chewiness in the crust.
Add toppings: Transfer shaped pizza to a parchment-lined peel or baking sheet. Brush with olive oil and sprinkle evenly with torn mozzarella. Arrange sliced figs over the cheese, then layer prosciutto on top. Season lightly with salt and pepper.
Bake: Carefully slide onto preheated stone (or place in oven if using a baking sheet). Bake for 12-15 minutes until crust is golden brown and edges are crisp. Watch closely toward the end — you want the mozzarella to bubble but not burn completely.
Finish: Remove from oven, immediately top with fresh arugula while pizza is still hot (the heat will slightly wilt it). Drizzle lightly with extra olive oil if desired. Let rest for 2 minutes before slicing.
