Use a hand-mixer to cream together the butter, granulated sugar, brown sugar, and oil in a medium bowl until the mixture is mostly smooth.
Beat in the egg, then beat in the lemon zest and vanilla extract.
Use a spatula or wooden spoon to stir the baking soda, baking powder, and salt into the batter.
Alternate adding the flour and the milk into the batter, stirring until just combined between each addition, until the batter is thick and smooth.
Divide the batter between the six rings of your prepared doughnut pan. Make sure that batter is evenly distributed around each ring and that there’s no batter covering the “hole” in the middle of the ring. Thump the doughnut pan a few times on your worksurface to release air bubbles.
Bake the doughnuts at 375F for about 12 minutes, or until a toothpick inserted into a ring comes out clean.
Remove the doughnuts pan from the oven and let cool for about 10 minutes.
Nudge the sides and centers of the doughnuts to make sure they release easily from the pan, then turn out the doughnuts onto a cooling rack. Let doughnuts cool completely before frosting.