Heat 1/2 cup of the oil in a medium skillet over medium-low heat.
Place the biscuits on a cutting board. Using a 1-inch round cookie cutter or shot glass, cut a hole in the center of each biscuit, reserving the extra dough for "holes."
Test the heat of the oil by dipping the edge of a doughnut in the pan. When the oil is hot enough, the edge will bubble. Place 4 of the doughnuts and holes in the skillet and cook until golden brown, 1 to 1 1/2 minutes per side. Transfer to a wire rack or paper towel-lined plate to drain. Add the remaining oil to the skillet, reheat, and cook the remaining doughnuts and holes.
In a large bowl, combine the sugar and cinnamon. Gently toss the warm doughnuts in the mixture a few at a time. Serve warm or at room temperature.