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Four hours before cooking, heat a wood-burning pizza oven. Alternately, heat a pizza stone in a 500 degree F oven for 1 hour.
For the pizza dough: In a stand mixer fitted with the dough hook, combine the yeast, lukewarm water, and 1/2 cup of the bread flour. Mix well and let sit for 30 minutes, or until bubbly like foamy beer.
In a separate medium bowl, whisk together the remaining 3 1/2 cups bread flour and the salt until evenly combined.
Add 3/4 cup cold water and the olive oil to the yeast mixture. Turn on the mixer and begin adding the flour mixture in increments, then mix for 5 minutes, until the dough starts creeping up the dough hook and detaching from the sides of the bowl.
Grease a large bowl with oil and transfer the dough to the bowl. Turn the dough to coat its surface with oil. Cover the bowl loosely with a tea towel. Let the dough rise until doubled in size, about 2 hours. Punch the dough down, then let rise another 1 hour.
Divide the dough into 4 equal pieces and form into discs. Sprinkle some all-purpose flour on your pizza peel. Using a rolling pin, stretch 1 piece of the dough to a very thin 10-inch round.
For the topping: Tear and scatter some of the Taleggio over the dough, leaving a 1-inch border. Tear and scatter some of the soppressata over the top, season with some salt and pepper and drizzle with some olive oil. Place the pizza on the oven floor and cook until the cheese is just melted and the dough is golden-brown and crisp, about 3 minutes. (If using a home oven with a pizza stone, increase the baking time by 2 minutes.)
Serve hot. Repeat with the 3 remaining dough balls.