Preheat your oven to 350°F (175°C). Prepare three 6-inch cake pans by greasing them lightly and lining each with parchment paper.
In the bowl of a stand mixer, combine the granulated sugar and lime zest. Rub together with your fingertips until the sugar becomes fragrant and resembles damp sand.
Add the cake flour, baking powder, and salt into the sugar mixture, blending on low speed.
Cut the butter into cubes and add it gradually to the dry ingredients, mixing until the texture is like fine sand.
Pour in the oil, milk, and vanilla extract, and mix just until everything is combined.
In a separate bowl, use a hand mixer to whip the egg whites until they reach medium peaks.
Gently fold the whipped egg whites into the batter using a rubber spatula, mixing carefully until just incorporated.
Evenly distribute the batter among the prepared cake pans.
Bake for approximately 22-25 minutes, or until the edges of the cakes pull slightly away from the pans. Transfer the cakes to a wire rack to cool completely.
Once cool, wrap each layer in plastic wrap and freeze until you’re ready to frost and decorate. (Frozen cake layers make for easier frosting.)