Pre-heat oven to 350°F. Grease an 8x8-inch cake pan.
In a small bowl, add the zest to the sugar and use the back of a small spoon to press the zest into the sugar. Continue until sugar is slightly damp and pale yellow.
In a medium bowl or measuring cup, whisk together the milk, egg whites and vanilla.
In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, lemon sugar, and baking powder. On the lowest speed, stir until combined, about 30 seconds.
Add the butter and increase speed to medium-low. Stir for 1 minute. Pour in the oil and continue stirring for 1 minute, until a smooth batter forms. Off the mixer and scrape down the sides.
Change the speed to medium and slowly pour in the milk mixture, in three additions, stirring for 20 seconds after each addition. Scrape down the sides after each addition as necessary.
Pour batter into the prepared cake pan. Bake for 20 – 22 minutes until a toothpick inserted comes out clean.
Let cake cool completely in the pan. Run the edge of a thin knife or an offset spatula along each side. Invert the cake onto a cooling rack, remove the pan, and let the inverted cake “dry” completely, about 1 hour.