Butter 2 8-inch cake pans and line with parchment paper.
In a medium bowl, whisk together the flours, baking powder, and salt.
In a large bowl, place the sugar and lemon zest. Using your hands, rub the lemon zest into the sugar for a minute or so until fragrant to release the lemon oil.
Add the butter to the bowl with the sugar and lemon zest. Using a stand or handheld mixer, cream the butter into the sugar-zest mixture, beating on high speed for about 3 minutes until smooth and creamy. Beat in the eggs one at a time, scraping down the bowl as necessary. Beat in the vanilla until well combined.
Add the flour mixture in three additions with the milk: flours, milk, flours, milk, and finish with the flours. Mix well on medium speed to make sure all the flours are incorporated.
Divide the batter evenly between the two pans. Cover and let rest for 10 minutes, then preheat oven to 350° F. Let cake batter continue to rest until oven is ready.
Place the cake in the oven and bake for about 30-35 minutes, checking at 25 minutes, until a tester inserted in the middle comes out clean. Remove from oven and place on a wire rack; let cake cool in pans for 10 minutes, then turn out onto the rack to cool completely.