Make the dough
Put the yeast in the bowl of a stand mixer, add the water, and whisk to dissolve. Add the salt, whisk to dissolve, and then add the flour. Attach the dough hook and mix on low speed until the flour is absorbed by the water, about 2 minutes. Slowly add up to 1-1/2 cups more flour as necessary, scraping down the sides of the bowl occasionally to ensure even kneading, until a firm, elastic, and tacky dough forms, 8 to 10 minutes; the dough shouldn’t stick to your fingers, but rather pull away clean like a Post-it note. (Alternatively, combine the dough ingredients as directed in a large bowl with a whisk and then a wooden spoon; knead by hand on a floured work surface, adding more flour as needed.)
Lightly coat a large bowl with olive oil. Transfer the dough to the bowl and turn to coat. Loosely cover with plastic wrap and let sit at room temperature for at least 6 and ideally 24 hours.
Shape and fry the crusts
Pour enough rice bran or canola oil into a 12-inch-wide, 9-quart heavy-duty pot to measure 2 inches deep. Clip a deep-fry thermometer to the pot, and heat the oil to 375°F.
On a lightly floured surface, divide the dough into 4 balls. Cover them with a damp kitchen towel.
Flour your hands, and working with one piece of dough at a time, press and stretch the dough into a 9-1/2-inch circle that’s about inch thick in the center, with edges that are slightly thicker. Use a fork to prick the dough evenly all over and all the way through.
Slowly lower the round of dough into the oil, releasing it away from you. Fry, turning once and using a wide metal spatula, slotted spoon, or tongs to occasionally submerge the dough until golden and slightly puffy, 1 to 2 minutes per side. The dough will blister and bubble; use the spatula (and/or tongs) to push the bubbles back down. If the center of the dough puffs up, lift up an edge with the tongs and release the air underneath. Transfer the crust to a large baking sheet or work surface. Repeat with the remaining dough, returning the oil to 375°F between batches.
Top and bake the pizzas
Position racks in the upper and lower thirds of the oven and set a large rimmed baking sheet upside down on each rack; heat the oven to 500°F.
In a large bowl, crush the tomatoes by hand into a slightly chunky sauce.
Spread 3 to 4 Tbs. of the tomato sauce over each crust, leaving a 1-inch border. Using a small spoon, scoop off pieces of the burrata and arrange over the tomato sauce. Tear the basil into small pieces and scatter on top.
Transfer two pizzas to each heated baking sheet and bake until the cheese melts, 3 to 5 minutes.
Drizzle each pizza with olive oil, season lightly with salt, and serve.
Photo: Eva Kolenko