For the pizza dough: In the bowl of a stand mixer, add 1/2 cup of the bread flour, the lukewarm water and the yeast. Mix well and let sit until bubbly, about 30 minutes.
Whisk together the remaining 3 1/2 cups bread flour and the salt in another bowl so it will be evenly distributed.
Once the yeast mixture is nice and bubbly and looks like foamy beer, add 3/4 cup cold water and the olive oil. Using a dough hook, turn the mixer on and add the flour in increments.
Mix the dough until it starts creeping up the dough hook and coming away from the sides of the bowl, about 5 minutes.
Place the dough in a large bowl lightly greased with olive oil. Turn to coat all sides of the dough with oil. Cover the bowl loosely with a clean tea towel. Let rise until doubled in size, about 2 hours.
Punch the dough down, and let rise another hour.
Divide the dough into 4 equal disks. Lightly flour a work surface. Using your fingers or heels of your hands (or a rolling pin, if you prefer) stretch the disks out to a 10- to 12-inch round. Poke your fingers in the dough a little to "dock" it. Repeat with the remaining 3 pizza dough disks.
Preheat the oven to 550 degrees F.
For the carpaccio: Chill the beef in the freezer for 10 minutes.
Thinly slice the beef against the grain with a very sharp knife. You should get close to 40 slices. You should only do this about 1 hour in advance of eating the meat, or it will discolor and become unappetizing. (You could also have your butcher do this for you on a professional meat slicer, but you will have to time it so that you are home in time to make the pizzas in the oven!) Store the carpaccio slices on waxed paper in one layer in the refrigerator until ready to use (up to 1 hour). If you stack the waxed paper layers, the bottom ones will discolor and become gray, so leaving them in one layer is best.
For the pizza: One at a time, drizzle each round of dough with olive oil, season lightly with salt and pepper, and sprinkle the surface with fresh rosemary.
Slide the dough onto a baking sheet into the oven and bake on the center rack until the sides are slightly charred and the pizza is cooked through, 10 to 12 minutes (less if you're using a grill!). Rotate the pizza in the oven or on the grill as necessary to keep it from burning.
Remove the pizza, set aside to cool off slightly and repeat with remaining pizzas.
To serve, use a pizza wheel or a sharp knife to cut each pizza into 8 to 10 slices. Lay 1 to 2 slices of the carpaccio over each slice. Sprinkle the micro arugula over the top of all of the slices and season with salt and pepper. Serve with a lemon wedge on the side. Repeat with the remaining pizzas.
To eat, squeeze the lemon wedge over the meat and enjoy immediately.
NotesCook's Notes: If hand slicing the carpaccio, slice them as thin as you can. If they're still not thin enough, lay the slices flat on a board, cover with plastic wrap and pound them lightly with a meat pounder or a wine bottle to flatten further.