Whisk together cake flour, all-purpose flour, sugar, and kosher salt in a bowl and set aside. Beat the egg until smooth with no streaks of white remaining.

In a small pot over low heat, melt butter, then whisk in the oil, milk, beaten egg, and vanilla. Cook, stirring constantly, until the mixture reaches 106°F, then pour into the bowl of a stand mixer.

Whisk the yeast into the warm milk mixture and let it bloom for 10 minutes until foamy.

Fit the mixer with the paddle attachment. Add half the flour mixture and mix on low to combine, then add the remaining flour and mix on low until fully incorporated.

Switch to the dough hook and mix on medium-high for approximately 12 minutes, until you can see strands pulling from the sides of the bowl into the center. The dough will not form a fully smooth ball.

Spray a large bowl and your hands with nonstick cooking spray. Scrape the dough into your hands, form into a ball, and place in the oiled bowl. Cover with plastic wrap and refrigerate for 30 minutes.

Turn the dough onto a lightly floured surface and roll gently to approximately ½ inch thickness. Cut donuts with a 3-inch cutter and holes with a 1-inch cutter directly on the baking sheet to prevent misshaping. Place on a parchment-lined baking sheet, spray the surface, cover with plastic wrap, and refrigerate for 2 hours.

Heat oil in a Dutch oven or fryer to 360°F, with at least 2 inches of oil depth. Keep oil between 350°F and 360°F throughout frying using a clip-on candy thermometer.

Let donuts rest at room temperature for 6 minutes before frying. Fry 3 to 5 donuts at a time, giving them room to puff. Once they rise to the surface, fry 60 seconds, flip, fry 45 seconds, flip again, and fry another 10 to 30 seconds until evenly golden brown. Test doneness with a cake tester: it should come out clean with no streaks of raw dough.

Transfer to a wire rack set over a sheet pan. Allow oil to drip off for 10 seconds, then roll in granulated sugar or dip in a simple powdered sugar glaze.