Prepare the Dough: In a medium mixing bowl, whisk together the sugar, baking powder, potato flour, baking soda, and salt. Add the melted butter, vanilla extract, egg yolks, and yogurt, stirring until thoroughly combined. Gradually incorporate the all-purpose flour until a soft dough forms. If measuring by cup, spoon the flour into the cup and level it off before sifting.
Knead and Roll the Dough: Transfer the dough to a generously floured work surface and knead gently until it is no longer sticky. Roll the dough to a thickness of ⅜ inch.
Cut the Doughnuts: Using a damp pastry brush, remove excess flour from the rolled dough. Cut out 1 ¾-inch rounds, arranging them on a baking sheet lined with paper towels. Brush off any remaining excess flour. Re-knead and roll scraps as needed to create additional doughnuts.
Create the Doughnut Holes: With a dampened fingertip, press an indentation into the center of each round, ensuring the dough does not stick to the paper towel. Flip the dough rounds over and repeat the process on the other side. Gently stretch the holes to widen them slightly.
Fry the Doughnuts: Heat 2 inches of refined coconut oil to 360°F. Fry the doughnuts until golden brown, then remove and drain on a paper towel-lined surface.