Fill a Dutch oven about halfway with veggie oil and heat to 375 degrees F. Line a sheet pan with paper towels and set aside.
Roll the pizza dough to about 1/4-inch thick on a floured work surface. Cut out as many 4-inch rounds as you can. Reroll the dough, if needed, to get 6 rounds total.
Combine the ricotta, confectioners' sugar and vanilla in a small bowl. Stir in the basil.
Dollop a scant tablespoon of filling in the center of each dough round and place a blackberry on top. Dab a little water around the edge of each dough round. Fold over to make half-moons and pinch well with your fingers to seal.
Add the pockets to the hot oil in batches and fry until golden and puffed, about 3 minutes. Remove to the lined sheet pan. Sprinkle on confectioners' sugar.