Position a rack in the center of the oven and preheat to 350 degrees F. Spray two 9-inch round cake pans with cooking spray and dust with flour.
Put the flour, baking powder and salt into the bowl of a food processor. Add the sugar and pulse a few times to combine evenly. Cut the butter into small pieces and add to the dry ingredients. Pulse until the mixture looks like coarse sand, with some pea-size bits of butter, about 5 times.
Whisk the milk, vanilla and eggs together in a liquid measuring cup. With the processor running, pour in the wet ingredients and process to make a smooth batter (this takes less than a minute). Divide the batter between the prepared pans and bake until a toothpick or small skewer inserted in the center comes out clean, about 35 minutes.
Cool the cakes on a rack for 15 minutes. Then run a knife around the edges of the pans and turn the cakes out onto a plate. Flip the cakes upright and cool completely on the rack, about 45 minutes. (This is important-icing a warm cake is not a happy thing.)