Pour enough canola or vegetable oil into a large Dutch oven to come 4 to 5 inches up the sides. Clip a deep-fry thermometer to the edge of the pan. Place over medium-high heat and bring to 365 degrees F.
Meanwhile, place the olive oil in a small saucepan over medium heat. Once warm, add the tomato puree and cook until bubbling and most water has evaporated, about 5 minutes. Remove from heat and set aside.
Dice the mozzarella and place in a strainer to remove any excess liquid. Set aside.
Dust a work surface with flour. Roll the pizza dough out until it is 1/4 inch thick. Using a 4- to 5-inch biscuit cutter, cut out rounds. Spread 1 to 2 teaspoons cooled tomato puree on one half of a round. Top with a basil leaf and 1 tablespoon mozzarella. Fold the other half of the round over to enclose the filling and pinch the edges to seal. Continue until all dough is filled and sealed. (At this point you can freeze the panzerotti for up to a month.)
Drop 2 to 3 panzerotti into the preheated oil and fry until golden, flipping them frequently, 3 to 4 minutes. Drain them on a plate lined with paper towels. Continue until all the panzerotti are cooked. Sprinkle with salt. Eat them hot!