Preheat the oven to 350°. Grease and flour two 9-inch round cake pans. In a medium bowl, whisk together the flour, baking powder and salt.
In a large bowl, using a mixer, beat the butter and sugar on medium speed until light and fluffy, 2 to 3 minutes. Scrape down the bowl. Add the eggs, one at a time, beating well after each addition. Beat in the vanilla. With the mixer on low speed, alternately add the dry ingredients and milk in three parts, mixing until just incorporated.
Divide the batter between the prepared pans and smooth the tops. Bake until a toothpick inserted in the center of each cake comes out clean, 35 to 40 minutes. Transfer the pans to a rack and let cool for 10 minutes. Run a knife around the pans to release the cakes and invert; let cool completely.
Slice the cakes into 2 layers each. Stack and frost with the vanilla frosting.