In a mixing bowl, whisk together the flour, baking powder, and salt. In another large bowl, whisk together the granulated sugar, whole egg, egg yolk, milk, vanilla, and 2 tablespoons oil together until combined. Add the dry ingredients to the wet and whisk to combine. The mixture should resemble thick pancake batter.
Transfer the batter to a piping bag fitted with a large (size 806) round tip. Pipe the batter in rings onto 4-inch parchment squares. You can refrigerate the piped doughnuts at this point until ready to fry, or fry them right away. If refrigerated, allow the dough to come to room temperature before frying.
In a large heavy-bottom pot, heat about 2 inches vegetable oil to 350°F. Fry the doughnuts in batches, slipping them still on the parchment into the oil. Fry about 3 minutes on the first side, until golden brown. Once the parchment releases from a doughnut, remove the parchment from the oil with tongs and discard. Flip the doughnuts and continue to fry about 1 minute longer on the second side, until golden. Place on a sheet pan fitted with a wire rack to drain.