Cut out eight 4-inch square parchment sheets. Brush both sides of parchment squares with vegetable oil. Arrange oiled squares on a large baking sheet.
Transfer chilled dough to a pastry bag (piping bag) fitted with a 16mm star tip. Pipe a large circle within the space of the parchment square. Apply even pressure when piping out the circle. At the end, simply release all pressure and pull the tip in the direction of the circle to cut dough. Don't worry about a little dough overlap.
Once all the cruller rings have been piped, place baking sheet in the freezer for 10-15 minutes while oil is heating. Do not cover dough. Simply place in freezer as is.
Attach candy thermometer to a large heavy bottomed pot (cast-iron or Dutch oven, preferred). Fill pot with 3-inches of oil*. Set over medium heat until an oil temperature of 360° F is achieved.
Remove cruller rings from freezer. Carefully place 2-3 cruller rings into the hot oil. Place the crullers into the oil upside down, with the parchment side up and the piped doughnut down.
Cook for 2 minutes and 30 seconds on the first side. Use kitchen tongs or kitchen chopsticks to carefully remove the parchment paper. Turn doughnuts over and cook for another 2 minutes. Turn doughnuts once more and cook for 45-60 seconds until golden in color.NOTE: Keep an eye on the thermometer and adjust heat as needed to maintain a constant cooking temperature.
Drain cooked doughnuts on a wire rack over a paper towel lined baking sheet.