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To prepare the dough: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. In a large bowl or the bowl of a stand mixer fitted with the flat beater attachment, combine the flour, water, yeast, olive oil, and salt. Mix until a shaggy, sticky dough forms without any dry, floury patches. This should take less than 1 minute. Scrape down the sides to gather the dough into a rough ball. Cover and set aside.
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After 30 minutes, use a bowl scraper or your wet hand to grab a section of dough from one side, lift it up, and press it down into the middle to seal. Repeat, turning the bowl 90° (a quarter turn) between each stretch, until the dough won’t elongate easily, about 8 to 12 times total. This process, which is called a bowl fold, replaces kneading.
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Cover the bowl and let the dough rest at room temperature, undisturbed, for 30 minutes. Perform a second bowl fold, then re-cover the bowl and let the dough rest, undisturbed, for at least 30 more minutes and up to 3 hours. This flexible dough will be fine with more or less time, but a longer rise will make shaping easier.
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Toward the end of the rise time, preheat the oven to 550°F (or 500°F, if that is your oven's highest setting) with a baking steel or stone on the center rack.
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Lightly flour the surface of the dough in the bowl, then turn the dough out onto a lightly floured work surface. Gently deflate it and shape it into a rough ball by pulling the edges into the center. Turn the dough seam-side down, cover, and let rest for at least 20 minutes and up to 1 hour, until it’s relaxed enough to stretch.
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Meanwhile, cut each piece of string cheese crosswise to make 4 stubby batons. Cut each baton lengthwise to make 2 planks, each with one flat side. In a small bowl, toss the string cheese with the pizza seasoning to coat evenly.
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To stretch the dough: Dust a sheet of parchment with flour. Transfer the ball of dough to the parchment and roll it into a 13" to 14" round, making the edges slightly thinner than the middle. Place the seasoned string cheese, flat-side down and end to end, in a circle around the perimeter of the dough about 1" from the edge. (You may have a couple pieces leftover depending on the size of your round.)
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Lightly brush the bare edge of the dough with water, then use floured fingers to fold the dough up and over the cheese, tucking it slightly underneath so that each piece of cheese is encased in dough. Press firmly to seal.
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Spread the pizza sauce evenly over the center of the pizza, then sprinkle the mozzarella cheese over. Brush the crust lightly with olive oil. Transfer the pizza, still on the parchment, to a baker’s peel or an inverted or rimless baking sheet. Trim any excess parchment around the dough to prevent it from burning.
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Use the baker’s peel to transfer the pizza, still on the parchment, to the preheated steel.
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Bake the pizza for 8 to 10 minutes, until the cheese is bubbling and brown. At the end of the bake time, switch the oven to broil for 1 to 2 minutes, watching carefully, to give the top more color, if desired. (If you do not have a broiler, you may need to add a couple minutes to the bake time.)
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Carefully remove the stuffed-crust pizza from the oven and transfer to a wire rack. Garnish with Parmesan and additional pizza seasoning and let cool slightly. Slice and enjoy warm, while the cheese crust is still melty.
