Heat oven to 350°. Butter and flour four 9″ cake pans; set aside. Whisk together flour, baking powder, and salt in a large bowl; set dry ingredients aside. Whisk together butter, sugar, milk, vanilla, and eggs in another bowl. Pour wet ingredients over dry ingredients, and using a whisk, stir together until just combined; let batter sit for 15 minutes. Stir batter again until smooth, and then divide half the batter among prepared pans; tilt cake pans around to let batter cover entire bottom. Bake cakes, rotating pans halfway through cooking, until barely browned, about 15 minutes. Let cakes cool for 20 minutes in pans, and then invert onto wire racks to cool completely. Clean and dry pans, and grease and flour again; divide remaining batter among pans, and repeat baking process.