An hour before you're ready to make your pizza, arrange the baking steel so that it's on the top rack of your oven and preheat the oven at 500°F. Take the dough out of the refrigerator and let it come to room temperature.
When you're ready to assemble your pizza, combine minced garlic, shallot, and rosemary with olive oil. Grate the mozzarella on the large holes of a box grater and set aside.
Plop your dough down on a generously floured clean countertop, and dust the surface with even more flour.
Pick the dough up in both hands with your fingertips and thumbs pointing down. The dough will begin to stretch toward the counter. Rotate the dough through your hands, letting the dough stretch downward as you go. Gravity will do most of the work, so be gentle and patient so you don't tear the dough.
When the dough is about 7-8" across, lay it flat on your counter and slide your hands under it with your palms against the counter. Stretch the dough on the backs of your hands by lifting them and and gently stretching them apart, rotating the dough between each stretch. If the dough gets too thin or you're worried it's going to tear, you can also lay it flat on the counter, gently pick up an edge of the dough with both hands, and hold it just a few inches off the counter to control the stretch as the dough gets thinner.
When the dough is thin in the center with a thicker crust and about 12-16" across, dust the pizza peel with a 50/50 blend of AP and semolina flour (about 1½ TBSP of each) and slide it under the dough. It helps if you have a second set of hands here so one person can lift the dough and the other can slide the peel under.
Work quickly to assemble the pizza so that the dough doesn't stick to the peel. Brush the surface with the olive oil mixture and top it with the grated mozzarella. Then arrange the prosciutto across the pizza, tearing it into a few big pieces with your hands. Let it drape and tent over itself so that lots of edges and corners are pointing up toward the broiler.
Right before you cook the pizza, turn the broiler on. Shake the pizza back and forth on the peel to make sure it hasn't stuck, then launch the pizza into the oven on the baking steel. Set a timer for 2 minutes.
Rotate the pizza and turn the broiler off. Cook for 2 more minutes. If it still looks pale, give it one more minute.
While the pizza is cooking, toss the arugula with the lemon juice, a drizzle of olive oil, salt, and pepper.
When the pizza is done, remove it from the oven. Top it with the arugula, cut it into slices and serve immediately.