Preheat oven to 350 degrees. Butter two 9-by-2-inch round cake pans; line the bottoms with parchment paper. Butter parchment, and dust with flour, tapping out excess; set aside. Into a medium bowl, sift together flours, baking powder, and salt; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and sugar until light and fluffy, 3 to 4 minutes, scraping down the sides of the bowl as needed. Beat in eggs, one at a time, then beat in vanilla.
With the mixer on low speed, add the flour mixture in three parts, alternating with the milk and beginning and ending with the flour; beat until combined after each addition.
Divide batter between the prepared pans, and smooth with an offset spatula. Bake, rotating the pans halfway through, until cakes are golden brown and a cake tester inserted into the center comes out clean, 30 to 35 minutes. Transfer pans to a wire rack to cool 20 minutes. Invert cakes onto the rack; peel off the parchment.⏱ 33 min
Reinvert cakes and let them cool completely, top sides up.