Preheat the oven to 350 degrees F. Butter two 9-inch-round cake pans and line the bottoms with parchment paper; butter the parchment and dust the pans with flour, tapping out the excess.
Whisk 3 cups flour, the baking powder and salt in a bowl until combined. Beat 2 sticks butter and the sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Reduce the mixer speed to medium; beat in the eggs, one at a time, scraping down the bowl as needed. Beat in the vanilla. (The mixture may look separated at this point.) Beat in the flour mixture in 3 batches, alternating with the milk, beginning and ending with flour, until just smooth.
Divide the batter between the prepared pans. Bake until the cakes are lightly golden on top and a toothpick inserted into the middle comes out clean, 30 to 35 minutes. Transfer to racks and let cool 10 minutes, then run a knife around the edge of the pans and turn the cakes out onto the racks to cool completely. Remove the parchment. Trim the tops of the cakes with a long serrated knife to make them level, if desired.