This recipe is not recommended for use with self-rising flour. I also use canola oil to fry but Mom used solid shortening. I don't remember oil being around then. Both work equally well.
If you don't have a doughnut cutter, cut with a No. 2 can (20 oz. can) that has been opened with a small can opener with a sharp blade to leave a sharp cutting edge. Use a pill bottle to make the center hole