Preheat the oven to 350°F/180°C. Grease a 9x13-inch baking dish with cooking spray.
In the bowl of a stand mixer, using the paddle attachment, stir together the flour, baking powder, salt, and granulated sugar.
In a separate bowl, whisk together the egg whites, coconut milk, dairy milk, vanilla, lime zest, and lime juice until combined. Whisk together until the mixture is smooth.
With the mixer on low, add the melted butter to the flour mixture a little at a time until combined. (This is the reverse creaming method. We are coating the flour with the fat (butter) to keep it from being able to create gluten strands. This gives us a very soft texture.)
Keep the mixer on low speed and slowly add the egg/milk mixture until completely combined. Bump up the speed to medium. Mix the batter for 2 minutes.
Transfer the batter into the prepared pan and spread out flat with an offset spatula. Bake for 35 to 40 minutes or until a toothpick inserted in the center comes out clean. (You should always bake to the INDICATOR, not the TIME, since all ovens are different! Use the time in any recipe as a guideline.)