Preheat oven to 350 degrees. Grease and flour 2 (8-inch) round cake pans, then line parchment paper. In a small bowl, whisk together milk, eggs, and vanilla. Set aside.
In the bowl of a standing mixer fit with the whisk attachment, beat together flour, sugar, baking powder, and salt.
With the mixer on medium-low speed, beat the butter in to the flour mixture one piece at a time. Continue beating until the mixture resembles moist crumbs, 1 to 3 minutes total.
Add the milk mixture to the mixer and beat until light and fluffy, 1 to 3 minutes longer. Stir by hand with a rubber spatula to make sure the batter is fully combined.
Divide the batter evenly between both pans, gently tapping the pans on the counter to settle the batter. Bake until a toothpick inserted in the center of the cake comes out dry with a few crumbs attached, 20 to 25 minutes. Rotate the pans halfway through baking time.
Cool the cakes in the pans for 10 minutes. Run a small sharp knife around the inside of each pan to loosen, then flip the cakes on to a wire rack. Peel off the parchment paper, then flip the cakes right side up to cool completely, about 2 hours.