Line a cupcake pan(s) with paper liners. You will need 12 or 13 lines. Preheat oven to 350°F.
In a large measuring cup with a pouring spout, whisk together the egg whites, milk, and vanilla. Set aside.
Using a stand mixer fitted with the paddle attachment, blend the flour, granulated sugar, baking powder and salt on low for about 30 seconds. With the mixer running, add the butter 1 piece at a time until incorporated and the mixture resembles moist crumbs.
Add all but about ½ cup of the egg mixture and blend just until combined. Increase the speed to medium-high and beat until light and fluffy, about 1 minute. Add the remaining ½cup of egg mixture in a slow stream. Scrape down the sides and beat on medium-high until well blended. Add the lime zest and juice and mix just until incorporated.
Scrape the batter into a large Ziplock bag and snip off one corner with scissors. Pipe the batter into the liners, dividing evenly between 12 or 13 cupcakes. I made 12 but they were a little big and over the edge, 13 would be about perfect. Bake in a 350°F oven for 18-20 minutes. Test the cupcakes with a toothpick inserted in the center. If it comes out clean, they are done.
Cool the cupcakes in the pan for 10 minutes. Transfer cupcakes to a rack and let cool completely before icing.