Melt butter over medium-heat in a small skillet or heavy-bottomed pot.
Continue to heat the butter, stirring periodically, until it turns brown (10-15 minutes). The butter will first boil with big bubbles (it's not done then). Then, the bubbles will get smaller and the whole thing will look more brown (that's when it's done).
Remove butter from heat and cool to room temperature.
In a small mixing bowl, whisk together cake flour, baking powder, baking soda, and salt.
In medium-sized bowl, whip egg whites until light and frothy.
Slowly mix sugar into the egg whites.
Mix cooled browned butter and vanilla into the wet mixture.
Alternately mix dry ingredients and milk into the wet mixture.
Fill cupcake liners 3/4 full.
Bake at 350 F for 20 minutes or until cupcakes bounce back when lightly touched.