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 it 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. On a work surface lightly dusted with flour, use your fingers or the heel of your hand (and a rolling pin, if you prefer) to stretch the discs until they're 13 inches in diameter. The disc should be very thin, less than 1/8 inch.
For the pizza: Place a pizza stone in the oven and preheat to 500 degrees F for 1 hour before baking. (Or preheat a wood-fired pizza oven to very hot, 4 hours before baking.)
In a food processor, puree the tomatoes until smooth. Season with salt to taste.
In a skillet, heat the olive oil and pancetta over medium heat and cook for 5 to 7 minutes, or until crisp.
Using a spoon, spread one-fourth of the pureed tomatoes on 1 disc of dough, leaving a 1-inch border all around. Sprinkle one-fourth of the mozzarella on top. Gently place the pizza on the pizza stone and bake for about 5 minutes, until golden with slight signs of charring. (If using a wood-fired oven, gently place the pizza on the oven floor, and bake for 3 minutes, or until golden and crisp.)
Remove the pizza from the oven and break an egg over the center of the pizza. Top with one-fourth of the cooked pancetta, return the pizza to the oven, and bake another 3 to 4 minutes (1 to 2 minutes if using a wood-fired oven), until the egg is cooked.
Remove the pizza from the oven, transfer to a cutting board, and top with one-fourth of the basil. Serve immediately. Repeat with the remaining 3 pizza discs. IMPORTANTE! If you're making pizza within the hour after forming the pizza dough discs, they can be kept at room temperature. Otherwise, place them on a baking sheet, cover with plastic wrap, and refrigerate for up to 12 hours before baking, as long as the dough doesn't rise too much, because acidity will creep in and the flavor will sour. Ideally, though, you should make the pizzas the same day you make the dough.
"Extra Virgin: Recipes & Love from Our Tuscan Kitchen" by Debi Mazar and Gabriele Corcos (c)Clarkson Potter 2014. Provided courtesy of Debi Mazar and Gabriele Corcos by arrangement with Clarkson Potter. All rights reserved.