For the dough: Combine the flour, yeast, and salt in the bowl of a mixer fitted with the paddle attachment. Combine the olive oil and 3/4 cup cold water in a spouted measuring cup. Pour combined liquids into the flour. Mix on low speed. Knead on medium-high speed until the dough forms a ball and is smooth and springy, 5 to 6 minutes.

Scrape the dough onto a floured surface, and knead until the dough comes together in a smooth ball. Coat a large bowl with olive oil, then add the dough. Cover, and let rise at room temperature until doubled in size, about 11/2 hours.

While the dough rises, make the filling: Combine the provola, prosciutto cotto, Grana Padano, and basil in a medium bowl, using your hands to toss and squeeze until the mixture comes together in a loose mass.

Once the dough has risen, punch it down. On a floured work surface, roll the dough to about 1/4 inch thick. Use a 41/2-inch round cutter to cut the dough into as many rounds as you can fit in. Divide the filling among the rounds, and wet the edges of each circle with your finger dipped in water or with a pastry brush. Press with your fingers to seal the edges
in a half-moon shape. Place on a parchment-lined baking sheet, roll out the dough scraps once more, and cut out more circles. (You’ll have about twelve in all.) Let them sit while the oil for frying heats, 10 to 15 minutes.

Heat 3 inches of vegetable oil in a large Dutch oven to 350 degrees Fahrenheit. Add half of the panzerotti and fry, turning occasionally, until they’re puffed and golden, about 5 minutes. Drain on a paper-towel-lined tray, and repeat with the second batch. Serve warm, with marinara.