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To make the dough: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Combine all the dough ingredients in the bowl of a stand mixer fitted with the dough hook. Mix on low speed for about 30 seconds. Stop and scrape the bowl, then mix on medium speed until a smooth dough forms that clears the sides of the bowl, 5 to 7 minutes.
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Remove the dough from the bowl and divide it into 2 equal pieces (about 215g each). Form into balls and place into separate lightly greased bowls. Cover and let rise for 60 to 75 minutes, or until puffy but not necessarily doubled in volume. Meanwhile, prepare the topping.
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To start the topping: In a medium bowl, combine the water and salt, stirring to dissolve the salt. Add the sliced onion and stir to submerge. Set aside, uncovered, at room temperature.
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To shape and fill the fugazzeta: Once the dough has risen, grease the bottom and sides of a 10" cast iron skillet with oil. Transfer one piece of dough to a clean (unfloured) work surface and use your fingertips or a rolling pin to gently stretch it into a 10" round, aiming to maintain an even 1/4" thickness from edge to edge. Transfer the dough to the skillet, pressing it so it completely covers the bottom of the pan.
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Shingle the slices of provolone evenly over the dough, leaving about a 1/2" border. In a medium bowl, combine the mozzarella, oregano, and red pepper flakes, then scatter the mixture evenly over the provolone, avoiding the dough border.
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Transfer the remaining piece of dough to the work surface and gently stretch it into a 10 1/2" to 11" round, again maintaining an even 1/4" thickness from edge to edge.
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Gently drape the second round of dough in the pan, aligning its edges with the first. Pinch the edges together to seal the filling inside. Cover and let rise at room temperature for 30 minutes.
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Meanwhile, preheat the oven to 450°F with a rack in the upper third.
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To finish the topping: Drain the onion, which should now look slightly plumped and translucent, and pat as dry as possible between paper towels (it may take a couple rounds) or a lint-free kitchen towel. Toss the onion with the dried oregano.
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By the end of the rising time, the fugazzeta should look just slightly puffed around the edges. Poke holes in the top with a fork to allow steam to escape. Scatter the onion mixture across the top of the fugazzeta, going all the way to the edges of the pan. Sprinkle evenly with the Parmesan cheese and drizzle with the olive oil.
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Bake for 25 to 30 minutes, until the edges are dark golden brown, the onion is charred in spots, and the Parmesan is golden brown. Let rest for 15 minutes before slicing; serve warm.
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Storage information: Store any leftover fugazzeta, well-wrapped, in the refrigerator for up to 3 days. Reheat individual servings in a 300°F oven for 5 to 10 minutes or in the microwave.