Make the dough: In a large bowl, mix the flour with the sugar, salt, and yeast. Make a well in the center of the flour and pour in the oil and 1 cup water. Stir with a wooden spoon until the dough comes together, and then cover with a kitchen towel. Let the dough stand at room temperature for 2 hours.
Divide the dough into 12 pieces and then shape each piece into a ball. Place the dough balls on a parchment paper-lined baking sheet and let stand for 1 hour.
Meanwhile, make the tomato sauce: In a 10-inch skillet, heat the oil over medium. Add the garlic, and cook, stirring, until golden, about 2 minutes. Add the tomatoes, and cook, stirring, until broken down into a sauce, about 20 minutes. Remove the garlic from the sauce, and then season with salt and pepper and stir in the basil.
On a lightly oiled work surface, roll each dough ball into a 5-inch disk. Place 1 heaping tablespoon of tomato sauce on one half of each disc, and then top with 2 tablespoons of the mozzarella. Fold the dough over the filling and press down with your thumb to seal it.
Pour enough oil to come 1 inch up the side of a deep 10-inch skillet, and attach a deep-fry thermometer to the side of the pan; heat the oil to 350°. Working with one panzerotti at a time, add to the oil and fry, turning once, until golden brown on both sides, about 4 to 5 minutes. Using a slotted spoon, remove each panzerotti from the oil and transfer to paper towels to drain. Repeat with the remaining panzerotti and serve while hot.