In a large bowl sift together flour, cake flour, baking powder, and salt. Set aside. In a pitcher, whisk together milk, half-and-half, and vanilla. Set aside.
Cream together butter and sugar in the bowl of a stand mixer until light and fluffy, about 1 minute. Add eggs one at time, mixing well and scraping the bowl after you add each egg.
Add ⅓ of the flour mixture. Add ½ of the milk. You add the dry ingredients in 3 parts, and the liquid in 2.
When everything has been added scrape the bowl one more time. Let the batter rest for 15 minutes.
Preheat the oven to 350 degrees and line two muffin tins with paper liners. Fill the paper liners three quarters of the way full. Bake for 20 minutes, or until a toothpick comes out clean. The cupcakes will be pale in color. Let cool for 5 minutes in the pan, then transfer to a rack to cool completely.
1½ cups (3 sticks) butter, room temperature
3 cups confectioners sugar, sifted
2 teaspoons vanilla extract
¼ teaspoon salt
Add the butter to the bowl of a stand mixer. Beat until the butter is pale, very smooth, and makes a slapping sound as it hits the sides of the bowl. This takes about 30 seconds if you butter is really soft, a bit longer if the butter is cooler.
Add the sugar one cup at a time. Mix until it is incorporated before you add the next cup.
When all of the sugar has been added, scrape the paddle and the sides of the bowl. Add the vanilla and salt, beat on low speed for 15 seconds.
Increase the speed to high (or just high enough so that it doesn’t make a mess). Beat until whipped, light, and perfectly smooth, about 5 minutes. Scrape the bowl once or twice. The frosting will look like it is getting really soft, but it will stiffen and increase in volume.
You can use this right away, or store in an airtight container in the fridge for up to one week.