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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.
Gently place the pizza on the oven floor and cook until golden and the dough is done, about 3 minutes. (If using a home oven with a pizza stone, increase the baking time by 5 minutes.)
Remove pizza to a cutting board, drizzle with some more olive oil, slice and serve immediately. Repeat with the remaining 3 dough balls and ingredients.