For the dough: Dissolve the yeast in the warm water in a large bowl; let stand for 5 minutes. Stir in 3 cups of the flour and the salt until smooth. Stir in an additional 2 cups flour; continue adding flour (up to 1/2 cup), 1 tablespoon at a time, stirring until the dough comes away from bowl but is still sticky.
Turn the dough out onto a floured work surface and knead it with lightly floured hands: Start by slapping the dough onto the counter, pulling it toward you with one hand and pushing it away with the other. Fold the dough back over itself (use a bench scraper or a wide knife to help scrape dough from work surface). Repeat until the dough is easier to handle, about 10 times. Finish by kneading normally until the dough is smooth, elastic and soft, but a little tacky, about 10 minutes.
Shape the dough into a ball and transfer it to a lightly oiled bowl; turn to coat. Cover with plastic and let rise in a warm place until doubled in volume, 3 hours. Press the dough with a finger to see if it's done; an indent should remain.
Turn out the dough onto a lightly floured surface and divide it into 4 to 6 balls. Place the balls on a lightly oiled baking sheet, cover gently with plastic wrap and let rest at room temperature until puffed and relaxed, about 45 minutes.
Meanwhile, make the sauce: Heat the oil in a large saucepan over high heat. Add the onions; cook until very soft, about 5 minutes. Add the garlic and chile flakes; cook 1 minute longer.
Add the tomatoes and their juices, bring to a boil, then reduce the heat to a strong simmer. Cook, stirring occasionally, until the tomatoes soften and the mixture thickens, about 30 minutes. Mash the tomatoes with a potato masher and continue cooking until thickened, about 15 minutes longer.
Stir in the oregano and season with salt and pepper. Cool before using on the pizza. (Makes 1 quart; freeze leftover sauce in a container with a tight-fitting lid for up to 1 month.)
For the pizza: Preheat a pizza oven to 625 degrees F, or a conventional oven as hot as it will go with a baking stone on or near the bottom of the oven.
Sprinkle a pizza peel with some cornmeal. Roll and stretch one of the dough balls into a thin oblong on a lightly floured surface and lay it on the peel. Spread the dough with about 1/2 cup tomato sauce, leaving a 1-inch border. Sprinkle the border with olive oil and season the whole pie with salt and pepper. Scatter 2 tablespoons of the Parmesan over the pizza and top with a few pieces of mozzarella -- it shouldn't cover the top. Slide the pizza into the pizza oven (or onto the baking stone) and cook until the crust is golden on the edges, blistered and crisp on the bottom, and the cheese is bubbling, 8 to 10 minutes, depending on oven temperature.
When the pizza still has a few minutes to go, heat some olive oil in a small nonstick skillet over medium-high heat. Slip in an egg and cook until about three-quarters done -- the white should be set enough to remove the egg from the pan but not completely firm. Season with salt and pepper. Use the peel to remove the pizza from the oven; slide the egg into the center of the pizza. Return the pizza to the oven and cook until the white is set and the yolk is still runny, 2 to 3 minutes. Transfer the pizza to a cutting board, top with torn basil, slice and serve. Repeat for the remaining pizzas.