In a medium bowl, combine the yeast with the sugar and 1/4 cup of lukewarm water and let stand until foamy, 5 minutes. Stir in 1/4 cup of the flour and let stand until very billowy, about 30 minutes.
Put the remaining 4 cups of flour and the salt in a standing mixer fitted with the dough hook. Add 1 1/2 cups of lukewarm water, the olive oil and the billowy dough and beat at medium speed until a firm but smooth and silky dough forms, about 10 minutes. Transfer the dough to a large oiled bowl, cover with a sheet of oiled plastic wrap and let stand until doubled in bulk, about 1 hour.
Punch down the dough, transfer it to a work surface and divide it into 8 pieces. Pinch the dough into balls and transfer them, rounded side up, to a floured baking sheet. Cover loosely with oiled plastic wrap and let rest for 20 minutes.
In a large, deep skillet, heat 1 inch of vegetable oil until shimmering. Roll out each ball of dough to an 8-inch round. Working with 1 round at a time, fry the bread over moderately high heat, turning once or twice, until the bread is puffed and deeply golden, 2 to 3 minutes; carefully spoon hot oil over any pale portions of the dough for even browning. Transfer the bread to a paper towellined baking sheet. Repeat with the remaining dough balls. Serve right away.
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